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- Verify solutions to linear equations with two variables.
- Graph lines by plotting points.
- Identify and graph horizontal and vertical lines.

A linear equation with two variablesAn equation with two variables that can be written in the standard form $ax+by=c$, where the real numbers *a* and *b* are not both zero. has standard form $ax+by=c$, where *a*, *b*, and *c* are real numbers and *a* and *b* are not both 0. Solutions to equations of this form are ordered pairs (*x*, *y*), where the coordinates, when substituted into the equation, produce a true statement.

**Example 1:** Determine whether (1, −2) and (−4, 1) are solutions to $6x-3y=12$.

**Solution:** Substitute the *x*- and *y*-values into the equation to determine whether the ordered pair produces a true statement.

Answer: (1, −2) is a solution, and (−4, 1) is not.

It is often the case that a linear equation is given in a form where one of the variables, usually *y*, is isolated. If this is the case, then we can check that an ordered pair is a solution by substituting in a value for one of the coordinates and simplifying to see if we obtain the other.

**Example 2:** Are $\left({\scriptscriptstyle \frac{1}{2}},\text{\hspace{0.17em}}-3\right)$ and $\left(-5,\text{\hspace{0.17em}}14\right)$ solutions to $y=2x-4$?

**Solution:** Substitute the *x*-values and simplify to see if the corresponding *y*-values are obtained.

Answer: $\left({\scriptscriptstyle \frac{1}{2}},\text{\hspace{0.17em}}-3\right)$ is a solution, and $\left(-5,\text{\hspace{0.17em}}14\right)$ is not.

**Try this!** Is (6, −1) a solution to $y=-\frac{2}{3}x+3$?

Answer: Yes

When given linear equations with two variables, we can solve for one of the variables, usually *y*, and obtain an equivalent equation as follows:

Written in this form, we can see that *y* depends on *x*. Here *x* is the independent variableThe variable that determines the values of other variables. Usually we think of the *x*-value as the independent variable. and *y* is the dependent variableThe variable whose value is determined by the value of the independent variable. Usually we think of the *y*-value as the dependent variable..

The linear equation $y=2x-4$ can be used to find ordered pair solutions. If we substitute any real number for *x*, then we can simplify to find the corresponding *y*-value. For example, if $x=3$, then $y=2(3)-4=6-4=2$, and we can form an ordered pair solution, (3, 2). Since there are infinitely many real numbers to choose for *x*, the linear equation has infinitely many ordered pair solutions (*x*, *y*).

**Example 3:** Find ordered pair solutions to the equation $5x-y=14$ with the given *x*-values {−2, −1, 0, 4, 6}.

**Solution:** First, solve for *y*.

Next, substitute the *x*-values in the equation $y=5x-14$ to find the corresponding *y*-values.

Answer: {(−2, −24), (−1, −19), (0, −14), (4, 6), (6, 16)}

In the previous example, certain *x*-values are given, but that is not always going to be the case. When treating *x* as the independent variable, we can choose any values for *x* and then substitute them into the equation to find the corresponding *y*-values. This method produces as many ordered pair solutions as we wish.

**Example 4:** Find five ordered pair solutions to $6x+2y=10$.

**Solution:** First, solve for *y*.

Next, choose any set of *x*-values. Usually we choose some negative values and some positive values. In this case, we will find the corresponding *y*-values when *x* is {−2, −1, 0, 1, 2}. Make the substitutions required to fill in the following table (often referred to as a t-chart):

Answer: {(−2, 11), (−1, 8), (0, 5), (1, 2), (2, −1)}. Since there are infinitely many ordered pair solutions, answers may vary depending on the choice of values for the independent variable.

**Try this!** Find five ordered pair solutions to $10x-2y=2$.

Answer: {(−2, −11), (−1, −6), (0, −1), (1, 4), (2, 9)} (*answers may vary*)

Since the solutions to linear equations are ordered pairs, they can be graphed using the rectangular coordinate system. The set of all solutions to a linear equation can be represented on a rectangular coordinate plane using a straight line connecting at least two points; this line is called its graphA point on the number line associated with a coordinate.. To illustrate this, plot five ordered pair solutions, {(−2, 11), (−1, 8), (0, 5), (1, 2), (2, −1)}, to the linear equation $6x+2y=10$.

Notice that the points are collinear; this will be the case for any linear equation. Draw a line through the points with a straightedge, and add arrows on either end to indicate that the graph extends indefinitely.

The resulting line represents all solutions to $6x+2y=10$, of which there are infinitely many. The steps for graphing lines by plotting points are outlined in the following example.

**Example 5:** Find five ordered pair solutions and graph: $10x-5y=10$.

**Solution:**

**Step 1:** Solve for *y*.

**Step2**: Choose at least two *x*-values and find the corresponding *y*-values. In this section, we will choose five real numbers to use as *x*-values. It is a good practice to choose 0 and some negative numbers, as well as some positive numbers.

Five ordered pair solutions are {(−2, −6), (−1, −4), (0, −2), (1, 0), (2, 2)}

**Step 3:** Choose an appropriate scale, plot the points, and draw a line through them using a straightedge. In this case, choose a scale where each tick mark on the *y*-axis represents 2 units because all the *y*-values are multiples of 2.

Answer:

It will not always be the case that *y* can be solved in terms of *x* with integer coefficients. In fact, the coefficients often turn out to be fractions.

**Example 6:** Find five ordered pair solutions and graph: $-5x+2y=10$.

**Solution:**

Remember that you can choose any real number for the independent variable *x*, so choose wisely here. Since the denominator of the coefficient of the variable *x* is 2, you can avoid fractions by choosing multiples of 2 for the *x*-values. In this case, choose the set of *x*-values {−6, −4, −2, 0, 2} and find the corresponding *y*-values.

Five solutions are {(−6, −10), (−4, −5), (−2, 0), (0, 5), (2, 10)}. Here we choose to scale the *x*-axis with multiples of 2 and the *y*-axis with multiples of 5.

Answer:

**Try this!** Find five ordered pair solutions and graph: $x+2y=6$.

Answer: {(−2, 4), (0, 3), (2, 2), (4, 1), (6, 0)}

We need to recognize by inspection linear equations that represent a vertical or horizontal line.

**Example 7:** Graph by plotting five points: $y=-2$.

**Solution:** Since the given equation does not have a variable *x*, we can rewrite it with a 0 coefficient for *x*.

Choose any five values for *x* and see that the corresponding *y*-value is always −2.

We now have five ordered pair solutions to plot {(−2, −2), (−1, −2), (0, −2), (1, −2), (2, −2)}.

Answer:

When the coefficient for the variable *x* is 0, the graph is a horizontal line. In general, the equation for a horizontal lineAny line whose equation can be written in the form *y* = *k*, where *k* is a real number. can be written in the form $y=k$, where *k* represents any real number.

**Example 8:** Graph by plotting five points: *x* = 3.

**Solution:** Since the given equation does not have a variable *y*, rewrite it with a 0 coefficient for *y*.

Choose any five values for *y* and see that the corresponding *x*-value is always 3.

We now have five ordered pair solutions to plot: {(3, −2), (3, −1), (3, 0), (3, 1), (3, 2)}.

Answer:

When the coefficient for the variable *y* is 0, the graph is a vertical line. In general, the equation for a vertical lineAny line whose equation can be written in the form *x* = *k*, where *k* is a real number. can be written as $x=k$, where *k* represents any real number.

To summarize, if *k* is a real number,

**Try this!** Graph $y=5$ and $x=-2$ on the same set of axes and determine where they intersect.

Answer: (−2, 5)

- Solutions to linear equations with two variables $ax+by=c$ are ordered pairs (
*x*,*y*), where the coordinates, when substituted into the equation, result in a true statement. - Linear equations with two variables have infinitely many ordered pair solutions. When the solutions are graphed, they are collinear.
- To find ordered pair solutions, choose values for the independent variable, usually
*x*, and substitute them in the equation to find the corresponding*y*-values. - To graph linear equations, determine at least two ordered pair solutions and draw a line through them with a straightedge.
- Horizontal lines are described by
*y*=*k*, where*k*is any real number. - Vertical lines are described by
*x*=*k*, where*k*is any real number.

Part A: Solutions to Linear Systems

*Determine whether the given point is a solution.*

1. $5x-2y=4$; (−1, 1)

2. $3x-4y=10$; (2, −1)

3. $-3x+y=-6$; (4, 6)

4. $-8x-y=24$; (−2, −3)

5. $-x+y=-7$; (5, −2)

6. $9x-3y=6$; (0, −2)

7. $\frac{1}{2}x+\frac{1}{3}y=-\frac{1}{6}$; (1, −2)

8. $\frac{3}{4}x-\frac{1}{2}y=-1$; (2, 1)

9. $4x-3y=1$; $\left({\scriptscriptstyle \frac{1}{2}},\text{\hspace{0.17em}}{\scriptscriptstyle \frac{1}{3}}\right)$

10. $-10x+2y=-\frac{9}{5}$; $\left({\scriptscriptstyle \frac{1}{5}},\text{\hspace{0.17em}}{\scriptscriptstyle \frac{1}{10}}\right)$

11. $y=\frac{1}{3}x+3$; (6, 3)

12. $y=-4x+1$; (−2, 9)

13. $y=\frac{2}{3}x-3$; (0, −3)

14. $y=-\frac{5}{8}x+1$; (8, −5)

15. $y=-\frac{1}{2}x+\frac{3}{4}$; $\left(-{\scriptscriptstyle \frac{1}{2}},\text{\hspace{0.17em}}1\right)$

16. $y=-\frac{1}{3}x-\frac{1}{2}$; $\left({\scriptscriptstyle \frac{1}{2}},\text{\hspace{0.17em}}-{\scriptscriptstyle \frac{2}{3}}\right)$

17. $y=2$; (−3, 2)

18. $y=4$; (4, −4)

19. $x=3$; (3, −3)

20. $x=0$; (1, 0)

*Find the ordered pair solutions given the set of **x**-values.*

21. $y=-2x+4$; {−2, 0, 2}

22. $y=\frac{1}{2}x-3$; {−4, 0, 4}

23. $y=-\frac{3}{4}x+\frac{1}{2}$; {−2, 0, 2}

24. $y=-3x+1$; {−1/2, 0, 1/2}

25. $y=-4$; {−3, 0, 3}

26. $y=\frac{1}{2}x+\frac{3}{4}$; {−1/4, 0, 1/4}

27. $2x-3y=1$; {0, 1, 2}

28. $3x-5y=-15$; {−5, 0, 5}

29. $\u2013x+y=3$; {−5, −1, 0}

30. $\frac{1}{2}x-\frac{1}{3}y=-4$; {−4, −2, 0}

31. $\frac{3}{5}x+\frac{1}{10}y=2$; {−15, −10, −5}

32. $x-y=0$; {10, 20, 30}

*Find the ordered pair solutions, given the set of **y**-values.*

33. $y=\frac{1}{2}x-1$; {−5, 0, 5}

34. $y=-\frac{3}{4}x+2$; {0, 2, 4}

35. $3x-2y=6$; {−3, −1, 0}

36. $-x+3y=4$; {−4, −2, 0}

37. $\frac{1}{3}x-\frac{1}{2}y=-4$; {−1, 0, 1}

38. $\frac{3}{5}x+\frac{1}{10}y=2$; {−20, −10, −5}

Part B: Graphing Lines

*Given the set of **x**-values {−2, −1, 0, 1, 2}, find the corresponding **y**-values and graph them.*

39. $y=x+1$

40. $y=-x+1$

41. $y=2x-1$

42. $y=-3x+2$

43. $y=5x-10$

44. $5x+y=15$

45. $3x-y=9$

46. $6x-3y=9$

47. $y=-5$

48. $y=3$

*Find at least five ordered pair solutions and graph.*

49. $y=2x-1$

50. $y=-5x+3$

51. $y=-4x+2$

52. $y=10x-20$

53. $y=-\frac{1}{2}x+2$

54. $y=\frac{1}{3}x-1$

55. $y=\frac{2}{3}x-6$

56. $y=-\frac{2}{3}x+2$

57. $y=x$

58. $y=-x$

59. $-2x+5y=-15$

60. $x+5y=5$

61. $6x-y=2$

62. $4x+y=12$

63. $-x+5y=0$

64. $x+2y=0$

65. $\frac{1}{10}x-y=3$

66. $\frac{3}{2}x+5y=30$

Part C: Horizontal and Vertical Lines

*Find at least five ordered pair solutions and graph them.*

67. $y=4$

68. $y=-10$

69. $x=4$

70. $x=-1$

71. $y=0$

72. $x=0$

73. $y=\frac{3}{4}$

74. $x=-\frac{5}{4}$

75. Graph the lines $y=-4$ and $x=2$ on the same set of axes. Where do they intersect?

76. Graph the lines $y=5$ and $x=-5$ on the same set of axes. Where do they intersect?

77. What is the equation that describes the *x*-axis?

78. What is the equation that describes the *y*-axis?

Part D: Mixed Practice

*Graph by plotting points.*

79. $y=-\frac{3}{5}x+6$

80. $y=\frac{3}{5}x-3$

81. $y=-3$

82. $x=-5$

83. $3x-2y=6$

84. $-2x+3y=-12$

Part E: Discussion Board Topics

85. Discuss the significance of the relationship between algebra and geometry in describing lines.

86. Give real-world examples relating two unknowns.

1: No

3: Yes

5: Yes

7: Yes

9: Yes

11: No

13: Yes

15: Yes

17: Yes

19: Yes

21: {(−2, 8), (0, 4), (2, 0)}

23: {(−2, 2), (0, 1/2), (2, −1)}

25: {(−3, −4), (0, −4), (3, −4)}

27: {(0, −1/3), (1, 1/3), (2, 1)}

29: {(−5, −2), (−1, 2), (0, 3)}

31: {(−15, 110), (−10, 80), (−5, 50)}

33: {(−8, −5), (2, 0), (12, 5)}

35: {(0, −3), (4/3, −1), (2, 0)}

37: {(−27/2, −1), (−12, 0), (−21/2, 1)}

39:

41:

43:

45:

47:

49:

51:

53:

55:

57:

59:

61:

63:

65:

67:

69:

71:

73:

75:

77: $y=0$

79:

81:

83: