By the end of this section, you will be able to: Be PreparedBefore we get started, take this readiness quiz.
Linear Expressions and Linear Equations with One VariableThe expressions in the Be Prepared section are examples of what we call linear expressions. Definition \(\PageIndex{1}\)1. An expression that can be written as \(A x+B\) with \(A\) and \(B\) real numbers, \(A\neq 0\), is called a linear expression (with one variable), or more specifically, a linear expression in \(x\). 2. An equation that can be written as \(A x+B=0\) with \(A\) and \(B\) real numbers, \(A\neq 0\), is called a linear equation (with one variable), or more specifically, a linear equation in \(x\). 3. A solution to a linear equation with one variable, say \(x\), is a number, say \(a\), that when substituted in for that variable yields a true statement. In this case we say that \(x=a\) is a solution. Be Prepared (3) can be written as \(0=3y+12\) by adding \(3y\) and subtracting \(8\) from both sides. So \(8-3y=20\) is a linear equation in \(y\). Notice that substituting \(y=1\), for example, into the equation gives \(\begin{aligned}8-3(1)&=20 \quad \text{ or }\\\quad 5&=20,\end{aligned}\) which is not true. However, substituting \(y=-4\) into the equation gives \(\begin{aligned}8-3(-4)&=20 \quad \text{ or }\\\quad 20&=20,\end{aligned}\) which is true. So \(y=-4\) is a solution to \(8-3y=20\), and \(y=1\) is not. In solving Be Prepared 3, we found that \(y=-4\) is the only solution. Introduction to Linear Expressions and Linear Equations with Two VariablesIn this section we will be looking at equations that have two variables, \(x\) and \(y\), and more than one solution. We want to represent the solutions in a picture. Consider the equation \(2x-3y=6\). The expression \(2x-3y\) is an example of a linear expression with two variables. We can evaluate the expressions on both sides of the equal sign for any particular choice of \(x\) and \(y\). For example, we can choose \(x=3\) and \(y=0\) and substitute them into the equation to get \(\begin{aligned}2(3) -3(0) &=6\quad \text{ or }\\\quad 6&=6,\end{aligned}\) which is a true statement. We can also choose \(x=-3\) and \(y=-4\) and substitute them into the equation to get \(\begin{aligned}2(-3) -3(-4) &=6\quad \text{ or }\\\quad 6&=6,\end{aligned}\) which is also true. We say that \(x=3\) and \(y=0\) is one solution, and \(x=-3\) and \(y=-4\) is another solution. We will actually see that equations like \(2x-3y=6\) have infinitely many solutions. Next we introduce what is needed to make a picture of the solutions. Plot Points on a Rectangular Coordinate SystemJust like maps use a grid system to identify locations, a grid system, or a rectangular coordinate system, is used in algebra to represent ordered pairs of numbers, and ultimately, to show a relationship between two variables. The rectangular coordinate system is also called the \(xy\)-plane or the “coordinate plane.” The rectangular coordinate system is formed by two intersecting number lines, one horizontal and one vertical. The horizontal number line is called the\(x\)-axis. The vertical number line is called the \(y\)-axis(note that in the context of an application these may take on different names). These axes divide a plane into four regions, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise. Definition \(\PageIndex{2}\)An ordered pair, \((x,y)\), gives the coordinates of a point in a rectangular coordinate system. The first number is the \(x\)-coordinate. The second number is the \(y\)-coordinate. The phrase “ordered pair” means that the order is important. For example, \((2,5)\) and \((5,2)\) are different points. What is the ordered pair of the point where the axes cross? At that point both coordinates are zero, so its ordered pair is \((0,0)\). The point \((0,0)\) has a special name. Definition \(\PageIndex{3}\)The point \((0,0)\) is called the origin. It is the point where the \(x\)-axis and \(y\)-axis intersect. We use the coordinates to locate a point on the \(xy\)-plane. Let’s plot the point \((1,3)\) as an example. First, locate the \(x\)-coordinate \(1\) on the \(x\)-axis and lightly sketch a vertical line through \(x=1\). Then, locate the \(y\)-coordinate \(3\) on the \(y\)-axis and sketch a horizontal line through \(y=3\). Now, find the point where these two lines meet -- that is the point with coordinates \((1,3)\). Notice that the vertical line through \(x=1\) and the horizontal line through \(y=3\) are not part of the graph. We just used them to help us locate the point \((1,3)\). When one of the coordinates is zero, the point lies on one of the axes. The graph below shows that the point \((0,4)\) is on the \(y\)-axis and the point \((−2,0)\) is on the \(x\)-axis. Points on the \(x\)- or \(y\)-axis
Example \(\PageIndex{4}\)Plot each point in the rectangular coordinate system and identify the quadrant in which the point is located: a. \((−5,4\)) b. \((−3,−4)\) c. \((2,−3)\) d. \((0,−1)\) e. \(\left(3,\dfrac{5}{2}\right)\) f. \((-2,3)\) SolutionThe first number of the coordinate pair is the \(x\)-coordinate, and the second number is the \(y\)-coordinate. To plot each point, sketch a vertical line through the \(x\)-coordinate and a horizontal line through the \(y\)-coordinate. Their intersection is the point. a. Since the \(x\)-coordinate is \(−5\), the point is to the left of the \(y\)-axis. Also, since the \(y\)-coordinate is \(4\), the point is above the \(x\)-axis. The point \((−5,4)\) is in Quadrant II. b. Since the \(x\)-coordinate is \(−3\), the point is to the left of the \(y\)-axis. Also, since the \(y\)-coordinate is \(−4\), the point is below the \(x\)-axis. The point \((−3,−4)\) is in Quadrant III. c. Since the \(x\)-coordinate is \(2\), the point is to the right of the \(y\)-axis. Since the \(y\)-coordinate is \(−3\), the point is below the \(x\)-axis. The point \((2,−3)\) is in Quadrant IV. d. Since the \(x\)-coordinate is \(0\), the point whose coordinates are \((0,−1)\) is on the \(y\)-axis. e. Since the \(x\)-coordinate is \(3\), the point is to the right of the \(y\)-axis. Since the \(y\)-coordinate is \(\dfrac{5}{2}\), the point is above the \(x\)-axis. (It may be helpful to write \(\dfrac{5}{2}\) as a mixed number, \(2\dfrac{1}{2}\), or decimal, \(2.5\), so that we know \(\dfrac{5}{2}\) is between \(2\) and \(3\).) The point \(\left(3,\dfrac{5}{2}\right)\) is in Quadrant I. f. Since the \(x\)-coordinate is \(-2\), the point is to the left of the \(y\)-axis. Since the \(y\)-coordinate is \(3\), the point is above the \(x\)-axis. The point \((-2,3)\) is in Quadrant II. The signs of the \(x\)-coordinate and \(y\)-coordinate affect the location of the points. We may have noticed some patterns as we graphed the points in the previous example. We can summarize sign patterns of the quadrants in this way: Quadrants
Linear Expressions and Linear Equations with Two VariablesUp to now, all the equations we have solved were equations with just one variable. In almost every case, when we solved the equation we got exactly one solution. But equations can have more than one variable. Equations with two variables may be of the form \(Ax+By=C\). An equation of this form is called a linear equation with two variables. Definition \(\PageIndex{6}\)1. An expression that can be written as \(A x+B y\) with \(A\) and \(B\) real numbers, not both zero, is called a linear expression (with two variables), or more specifically, a linear expression in \(x\) and \(y\). 2. An equation that can be written as \(A x+By=C\) with \(A\) and \(B\) real numbers, not both zero, is called a linear equation (with two variables), or more specifically, a linear equation in \(x\) and \(y\). Here is an example of a linear equation with two variables, \(x\) and \(y\). \(x+4y=8\) This equation is in the form \(Ax+By=C\) with \(A=1\), \(B=4\), and \(C=8\). The equation \(y=−3x+5\) is also a linear equation with two variables, \(x\) and \(y\), but it does not appear to be in the form \(Ax+By=C\). We can rewrite it in \(Ax+By=C\) form in the following way.
By rewriting \(y=−3x+5\) as \(3x+y=5\), we can easily see that it is a linear equation with two variables because it is of the form \(Ax+By=C\) with \(A=3\), \(B=1\), and \(C=5\). When an equation is in the form \(Ax+By=C\), we say it is in standard form of a linear equation. Definition \(\PageIndex{7}\)A linear equation with two variables, \(x\) and \(y\), is in standard form when it is written as \(Ax+By=C\). Most people prefer to have \(A\), \(B\), and \(C\) be integers and \(A \geq 0\) when writing a linear equation in standard form, although it is not strictly necessary. Linear equations with two variables have infinitely many solutions. For example, if \(A \neq 0\), for every number that is substituted for \(x\) there is a corresponding \(y\)-value. This pair of values is a solution to the linear equation and is represented by the ordered pair \((x,y)\). When we substitute these values of \(x\) and \(y\) into the equation, the result is a true statement, because the value on the left side is equal to the value on the right side. Definition \(\PageIndex{8}\)An ordered pair \((p,q)\) is a solution of the linear equation \(Ax+By=C\), if the equation is a true statement when the \(x\)- and \(y\)-coordinates of the ordered pair, \(p\) and \(q\), respectively, are substituted into the equation. We can also say in this case that \((x,y)=(p,q)\) is a solution, or \(x=p\) and \(y=q\) is a solution. We can represent the solutions as points in the rectangular coordinate system. The points will line up perfectly in a straight line. We use a straight-edge to draw this line, and put arrows on the ends of each side of the line to indicate that the line continues in both directions. A graph is a visual representation of all the solutions of the equation. It is an example of the saying, “A picture is worth a thousand words.” The line (with the arrows) shows us all the solutions to that equation. Every point on the line corresponds a solution of the equation. And, every solution of this equation corresponds to a point on this line. This line is called the graph of the equation. Points not on the line do not correspond to solutions! We may say, as is common, then that the points on the line are the solutions. Definition \(\PageIndex{9}\)The graph of the linear equation \(Ax+By=C\) is the collection of all solutions \((x,y)\). We can represent the graph on the coordinate plane. The representation is a straight line so that
As universally accepted, a representation is also called a graph of the linear equation. Example \(\PageIndex{10}\)The graph of \(y=2x−3\) is shown below. For each ordered pair, \(A(0,−3),\qquad B(3,3), \qquad C(2,−3),\qquad\text{ and } \qquad D(−1,−5),\) decide: a. is the ordered pair a solution to the equation? b. is the point on the line? SolutionSubstitute the \(x\)- and
\(y\)-values into the equation to check if the ordered pair is a solution to the equation.
b. Plot the points \((0,−3)\), \((3,3)\), \((2,−3)\), and \((−1,−5)\). The points \((0,3)\), \((3,−3)\), and \((−1,−5)\) are on the line \(y=2x−3\), and the point \((2,−3)\) is not on the line. Try It \(\PageIndex{11}\)The graph of \(y=3x−1\) is shown below. For each ordered pair, \(A(0,−1)\qquad\text{ and } \qquad B(2,5),\) decide: a. is the ordered pair a solution to the equation? a. Both pairs are solutions. b. Both pairs are on the line. Try It \(\PageIndex{12}\)The graph of \(y=3x−1\) is shown below. For each ordered pair, \(A(3,−1)\qquad\text{ and } \qquad B(−1,−4),\) decide: a. Is the ordered pair a solution to the equation? a. \(A\) is a solution. \(B\) is not a solution. b. \(A\) is on the line. \(B\) is not on the line. Graph a Linear Equation by Plotting PointsThere are several methods that can be used to graph a linear equation. The first method we will use is called plotting points. We find three points whose coordinates are solutions to the equation and then plot them in a rectangular coordinate system. By connecting these points in a line, we have the graph of the linear equation. While two points are enough to determine a line, using three points helps us detect errors. Example \(\PageIndex{13}\)Graph the equation \(y=2x+1\) by plotting points. SolutionTry It \(\PageIndex{14}\)Graph the equation \(y=2x−3\) by plotting points. AnswerTry It \(\PageIndex{15}\)Graph the equation \(y=−2x+4\) by plotting points. AnswerThe steps to take when graphing a linear equation by plotting points are summarized here. Graph a linear equation by plotting points
It is true that it only takes two points to determine a line, but it is a good habit to use three points. If we only plot two points and one of them is incorrect, we can still draw a line but it will not represent the solutions to the equation. It will be the wrong line. If we use three points, and one is incorrect, the points will not line up. This tells us something is wrong and we need to check our work. Look at the difference between these illustrations. When an equation includes a fraction as the coefficient of \(x\), we can still substitute any numbers for \(x\). But the arithmetic is easier if we make “good” choices for the values of \(x\). This way we will avoid fractional answers, which are hard to plot precisely. Example \(\PageIndex{16}\)Graph the equation \(y=\dfrac{1}{2}x+3\). SolutionFind three points that are solutions to the equation. Since this equation has the fraction \(\dfrac{1}{2}\) as a coefficient of \(x\), we will choose values of \(x\) carefully. We will use zero as one choice and multiples of 2 for the other choices. Why are multiples of two a good choice for values of \(x\)? By choosing multiples of 2 the multiplication by \(\dfrac{1}{2}\) simplifies to a whole number.
We organize the three solutions in a table.
Plot the points, check that they line up, and draw the line. Try It \(\PageIndex{17}\)Graph the equation \(y=\dfrac{1}{3}x−1\). AnswerTry It \(\PageIndex{18}\)Graph the equation \(y=\dfrac{1}{4}x+2\). AnswerGraph Vertical and Horizontal LinesSome linear equations have only one variable. They may have just \(x\) and no \(y\), or just \(y\) without an \(x\). This changes how we make a table of values to get the points to plot. Let’s consider the equation \(x=−3\). This equation has only one variable, \(x\). The equation says that the \(x\)-coordinate of any solutionis equal to \(−3\), so its value does not depend on the \(y\)-coordinate. No matter what is the value of the \(y\)-coordinate, the value of the \(x\)-coordinate is always \(−3\). So to make a table of values, write \(−3\) in for all the \(x\)-coordinates. Then choose any values for the \(y\)-coordinate. Since the \(x\)-coordinate does not depend on the \(y\)-coordinate, we can choose any numbers we like. But to fit the points on our coordinate graph, we’ll use 1, 2, and 3 for the \(y\)-coordinates.
Plot the points from the table and connect them with a straight line. Notice that we have graphed a vertical line. What if the equation has \(y\) but no \(x\)? Let’s graph the equation \(y=4\). This time the \(y\)-coordinate of any solution is 4, so for this equation, the \(y\)-coordinate of the solution does not depend on the \(x\)-coordinate. Fill in 4 for all the \(y\)-coordinates in the table and then choose any values for \(x\)-coordinates. We will use 0, 2, and 4 for the \(x\)-coordinates.
In this figure, we have graphed a horizontal line passing through the \(y\)-axis at 4. Definition \(\PageIndex{19}\)1. A vertical line is the graph of an equation (with two variables \(x\) and \(y\)) of the form \(x=a\). The line passes through the \(x\)-axis at \((a,0)\). 2. A horizontal line is the graph of an equation (with two variables \(x\) and \(y\)) of the form \(y=b\). The line passes through the \(y\)-axis at \((0,b)\). Example \(\PageIndex{20}\)Graph: a. \(x=2\) b. \(y=−1\) Solutiona. The equation has only one variable, \(x\), and \(x\) is always equal to 2. We create a table where the \(x\)-coordinate is always 2 and then put in any values for the \(y\)-coordinate. The graph is a vertical line passing through the \(x\)-axis at 2.
b. Similarly, the equation \(y=−1\) has only one variable, \(y\). The value of the \(y\)-coordinate of any solution is \(-1\). All the ordered pairs in the next table have the same \(y\)-coordinate. The graph is a horizontal line passing through the \(y\)-axis at \(−1\).
Try It \(\PageIndex{21}\)Graph: a. \(x=5\) b. \(y=−4\) Answera. b. Try It \(\PageIndex{22}\)Graph: a. \(x=−2\) b. \(y=3\) Answera. b. What is the difference between the equations \(y=4x\) and \(y=4\)? The equation \(y=4x\) has both \(x\) and \(y\). The value of the \(y\)-coordinate of a solution depends on the value of the \(x\)-coordinate, so the \(y\)-coordinate changes according to the value of the \(x\)-coordinate. The equation \(y=4\) has only one variable. The value of \(y\)-coordinate of any solution is \(4\), it does not depend on the value of the \(x\)-coordinate. Notice, in the graph, the equation \(y=4x\) gives a slanted line, while \(y=4\) gives a horizontal line. Example \(\PageIndex{23}\)Graph \(y=−3x\) and \(y=−3\) in the same rectangular coordinate system. SolutionWe notice that the first equation has the variable \(x\), while the second does not. We make a table of points for each equation and then graph the lines. The two graphs are shown. Try It \(\PageIndex{24}\)Graph \(y=−4x\) and \(y=−4\) in the same rectangular coordinate system. AnswerTry It \(\PageIndex{25}\)Graph \(y=3\) and \(y=3x\) in the same rectangular coordinate system. AnswerFind \(x\)- and \(y\)-interceptsEvery linear equation can be represented by a line. We have seen that when graphing a line by plotting points, we can use any three solutions to graph. This means that two people graphing the line might use different sets of three points. At first glance, their two lines might not appear to be the same, since they would have different points labeled. But if all the work was done correctly, the lines should be exactly the same. One way to recognize that they are indeed the same line is to look at where the line intersects the \(x\)-axis and the \(y\)-axis. These points are called the intercepts of a line. Definition \(\PageIndex{26}\)The points where a graph crosses the \(x\)-axis and the \(y\)-axis are called the intercepts of the graph. Let’s look at the graphs of the lines. First, notice where each of these lines crosses the \(x\)-axis. Now, let’s look at the points where these lines cross the \(y\)-axis.
Is there a pattern? For each line, the \(y\)-coordinate of the point where the line crosses the \(x\)-axis is zero. The point where the line crosses the \(x\)-axis has the form \((a,0)\) and is called the \(x\)-intercept of the line. The \(x\)-intercept occurs when \(y\) is zero. In each line, the \(x\)-coordinate of the point where the line crosses the \(y\)-axis is zero. The point where the line crosses the \(y\)-axis has the form \((0,b)\) and is called the \(y\)-intercept of the line. The \(y\)-intercept occurs when \(x\) is zero. Definition \(\PageIndex{27}\)
Example \(\PageIndex{28}\)Find the \(x\)- and \(y\)-intercepts on each graph shown. a. The graph crosses the \(x\)-axis at the point \((4,0)\). The \(x\)-intercept is \((4,0)\). b. The graph crosses the \(x\)-axis at the point \((2,0)\). The \(x\)-intercept is \((2,0)\). c. The graph crosses the \(x\)-axis at the point \((−5,0)\). The \(x\)-intercept is \((−5,0)\). Try It \(\PageIndex{29}\)Find the \(x\)- and \(y\)-intercepts on the graph. The
\(x\)-intercept is \((2,0)\). Try It \(\PageIndex{30}\)Find the \(x\)- and \(y\)-intercepts on the graph. The \(x\)-intercept is \((3,0)\). Recognizing that the \(x\)-intercept occurs when \(y\) is zero and that the \(y\)-intercept occurs when \(x\) is zero gives us a method to find the intercepts of a line from its equation. To find the \(x\)-intercept, let \(y=0\) and solve for \(x\). To find the \(y\)-intercept, let \(x=0\) and solve for \(y\). Intercepts from the equation of a lineTo find:
Example \(\PageIndex{31}\)Find the intercepts of the graph of \(2x+y=8\). SolutionWe will let \(y=0\) to find the \(x\)-intercept, and let \(x=0\) to find the \(y\)-intercept. We will fill in a table, which reminds us of what we need to find. Let's find the \(x\)-intercept first.
Now, the \(y\)-intercept.
The intercepts are the points \((4,0)\) and \((0,8)\) as shown in the table.
The \(x\)-intercept is \((4,0)\). The \(y\)-intercept is \((0,8)\). Try It \(\PageIndex{32}\)Find the intercepts of the graph of \(3x+y=12\). AnswerThe \(x\)-intercept is
\((4,0)\). Try It \(\PageIndex{33}\)Find the intercepts of the graph of \(x+4y=8\). AnswerThe \(x\)-intercept is \((8,0)\). Graph a Line Using the InterceptsTo graph a linear equation by plotting points, we need to find three points whose coordinates are solutions to the equation. We can use the \(x\)- and \(y\)- intercepts as two of our three points. Find the intercepts, and then find a third point to ensure accuracy. Make sure the points line up—then draw the line. This method is often the quickest way to graph a line. Example \(\PageIndex{34}\)Graph \(–x+2y=6\) using the intercepts. SolutionTry It \(\PageIndex{35}\)Graph \(x-2y=4\) using the intercepts. AnswerTry It \(\PageIndex{36}\)Graph \(–x+3y=6\) using the intercepts. AnswerWhen the line passes through the origin, the \(x\)-intercept and the \(y\)-intercept are the same point. Example \(\PageIndex{37}\)Graph \(y=5x\) using the intercepts. Solution
This line has only one intercept. It is the point \((0,0)\).
The resulting three points are summarized in the table.
Plot the three points, check that they line up, and draw the line. Try It \(\PageIndex{38}\)Graph \(y=4x\) using the intercepts. AnswerTry It \(\PageIndex{39}\)Graph \(y=−x\) using the intercepts. AnswerKey Concepts
Glossaryhorizontal lineA horizontal line is the graph of an equation of the form \(y=b\). The line passes through the \(y\)-axis at \((0,b)\).intercepts of a lineThe points where a line crosses the \(x\)-axis and the \(y\)-axis are called the intercepts of the line.linear equationAn equation of the form \(Ax+By=C\), where \(A\) and \(B\) are not both zero, is called a linear equation with two variables.ordered pairAn ordered pair, \((x,y)\) gives the coordinates of a point in a rectangular coordinate system. The first number is the \(x\)-coordinate. The second number is the \(y\)-coordinate.originThe point \((0,0)\) is called the origin. It is the point where the \(x\)-axis and \(y\)-axis intersect.solution of a linear equation with two variablesAn ordered pair \((x,y)\) is a solution of the linear equation \(Ax+By=C\), if the equation is a true statement when the \(x\)- and \(y\)-values of the ordered pair are substituted into the equation.standard form of a linear equationA linear equation is in standard form when it is written \(Ax+By=C\).vertical lineA vertical line is the graph of an equation of the form \(x=a\). The line passes through the \(x\)-axis at \((a,0)\).Practice Makes PerfectPlot Points in a Rectangular Coordinate System In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located. 1. a. \((−4,2)\) b. \((−1,−2)\) c.
\((3,−5)\) d. \((−3,0)\) 2. a. \((−2,−3)\) b. \((3,−3)\) c. \((−4,1)\) d. \((4,−1)\) 3. a.
\((3,−1)\) b. \((−3,1)\) c. \((−2,0)\) d. \((−4,−3)\) 4. a. \((−1,1)\) b. \((−2,−1)\) c. \((2,0)\) d. \((1,−4)\) In the following exercises, for each ordered pair, decide a. is the ordered pair a solution to the equation? b. is the point on the line? 5. \(y=x+2\); A: \((0,2)\); B: \((1,2)\); C: \((−1,1)\); D: \((−3,−1)\). a. A: yes, B: no, C: yes, D: yes b. A: yes, B: no, C: yes, D: yes 6. \(y=x−4\); A: \((0,−4)\); B: \((3,−1)\); C: \((2,2)\); D: \((1,−5)\). 7. \(y=\dfrac{1}{2}x−3\); a. A: yes, B: yes, C: yes, D: no b. A: yes, B: yes, C: yes, D: no 8. \(y=\dfrac{1}{3}x+2\); Graph a Linear Equation by Plotting Points In the following exercises, graph by plotting points. 9. \(y=x+2\) Answer10. \(y=x−3\) 11. \(y=3x−1\) Answer12. \(y=−2x+2\) 13. \(y=−x−3\) Answer14. \(y=−x−2\) 15. \(y=2x\) Answer16. \(y=−2x\) 17. \(y=\dfrac{1}{2}x+2\) Answer18. \(y=\dfrac{1}{3}x−1\) 19. \(y=\dfrac{4}{3}x−5\) Answer20. \(y=\dfrac{3}{2}x−3\) 21. \(y=−\dfrac{2}{5}x+1\) Answer22. \(y=−\dfrac{4}{5}x−1\) 23. \(y=−\dfrac{3}{2}x+2\) Answer24. \(y=−\dfrac{5}{3}x+4\) Graph Vertical and Horizontal lines In the following exercises, graph each equation. 25. a. \(x=4\) b. \(y=3\) Answera. b. 26. a. \(x=3\) b. \(y=1\) 27. a. \(x=−2\) b. \(y=−5\) Answera. b. 28. a. \(x=−5\) b. \(y=−2\) In the following exercises, graph each pair of equations in the same rectangular coordinate system. 29. \(y=2x\) and \(y=2\) Answer30. \(y=5x\) and \(y=5\) 31. \(y=−\dfrac{1}{2}x\) and \(y=−\dfrac{1}{2}\) Answer32. \(y=−\dfrac{1}{3}x\) and \(y=−\dfrac{1}{3}\) Find x- and y-Intercepts In the following exercises, find the x- and y-intercepts on each graph. 33. \((3,0),(0,3)\) 34. 35. \((5,0),(0,−5)\) 36. In the following exercises, find the intercepts for each equation. 37. \(x−y=5\) Answer\(x\)-int: \((5,0)\), \(y\)-int: \((0,−5)\) 38. \(x−y=−4\) 39. \(3x+y=6\) Answer\(x\)-int: \((2,0)\), \(y\)-int: \((0,6)\) 40. \(x−2y=8\) 41. \(4x−y=8\) Answer\(x\)-int: \((2,0)\), \(y\)-int: \((0,−8)\) 42. \(5x−y=5\) 43. \(2x+5y=10\) Answer\(x\)-int: \((5,0)\), \(y\)-int: \((0,2)\) 44. \(3x−2y=12\) Graph a Line Using the Intercepts In the following exercises, graph using the intercepts. 45. \(−x+4y=8\) Answer46. \(x+2y=4\) 47. \(x+y=−3\) Answer48. \(x−y=−4\) 49. \(4x+y=4\) Answer50. \(3x+y=3\) 51. \(3x−y=−6\) Answer52. \(2x−y=−8\) 53. \(2x+4y=12\) Answer54. \(3x−2y=6\) 55. \(2x−5y=−20\) Answer56. \(3x−4y=−12\) 57. \(y=−2x\) Answer58. \(y=5x\) 59. \(y=x\) Answer60. \(y=−x\) Mixed Practice In the following exercises, graph each equation. 61. \(y=\dfrac{3}{2}x\) Answer62. \(y=−\dfrac{2}{3}x\) 63. \(y=−\dfrac{1}{2}x+3\) Answer64. \(y=\dfrac{1}{4}x−2\) 65. \(4x+y=2\) Answer66. \(5x+2y=10\) 67. \(y=−1\) Answer68. \(x=3\) Writing Exercises69. Explain how you would choose three x-values to make a table to graph the line \(y=\dfrac{1}{5}x−2\). AnswerAnswers will vary. 70. What is the difference between the equations of a vertical and a horizontal line? 71. Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation \(4x+y=−4\)? Why? AnswerAnswers will vary. 72. Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation \(y=\dfrac{2}{3}x−2\)? Why? Self Checka. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. b. If most of your checks were: Confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific. With some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved? No, I don’t get it. This is a warning sign and you must address it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need. |