Solutions to Equations with Two Variables

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 $\left( \frac{1}{2},-3\right)$ and $\left(-5,14\right)$ solutions to $y=2x-4$?

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

Answer: $\left( \frac{1}{2},-3\right)$ is a solution, and $\left(-5,14\right)$ 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)

Graph by Plotting Points

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)}

Horizontal and Vertical Lines

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)

Answers

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)}

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77: $y=0$

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