Graphing Linear Equations in <var>x</var> and <var>y</var>

Graph $x + y = 5$ .

The equation $x + y = 5$ is linear.

Find two solutions of $x + y = 5$ . Since there are infinitely many solutions, there are many possibilities.

Substitute 7 for x in $x + y = 5$ , and solve for y.

$x + y = 5$ $7 + y = 5$ $y = - 2$

$(7, - 2)$ is a solution.

Substitute 1 for x in $x + y = 5$ , and solve for y.

$x + y = 5$ $1 + y = 5$ $y = 4$

$(1, 4)$ is a solution.
Plot the solutions $(7, - 2)$ and $(1, 4)$ .
Draw the line through the two points $(7, - 2)$ and $(1, 4)$ .

Any two solutions of the linear equation can be used to graph it. In the next example,

x + y = 5

will be graphed again using solutions different from the solutions used in the previous example. The final answer is the same.

Graph $x + y = 5$ .

The equation $x + y = 5$ is linear.

Find two solutions of $x + y = 5$ .

Substitute 2 for y in $x + y = 5$ , and solve for x.

$x + y = 5$ $x + 2 = 5$ $x = 3$

$(3, 2)$ is a solution.

Substitute $- 4$ for x in $x + y = 5$ , and solve for y.

$x + y = 5$ $- 4 + y = 5$ $y = 9$

$(- 4, 9)$ is a solution.
Plot the solutions $(3, 2)$ and $(- 4, 9)$ .
Draw the line through the two points $(3, 2)$ and $(- 4, 9)$ .

Finding three solutions instead of two solutions may catch any errors made. If the three points are not on a line, an error has been made.

Graph $3 x + 2 y = 12$ .

Find three solutions to $3 x + 2 y = 12$ .

Substitute 0 for y in $3 x + 2 y = 12$ , and solve for x.

$3 x + 2 y = 12$ $3 x + 2 (0) = 12$ $3 x = 12$ $x = 4$

$(4, 0)$ is a solution.

Substitute 0 for x in $3 x + 2 y = 12$ , and solve for y.

$3 x + 2 y = 12$ $3 (0) + 2 y = 12$ $2 y = 12$ $y = 6$

$(0, 6)$ is a solution.

Substitute $- 2$ for x in $3 x + 2 y = 12$ , and solve for y.

$3 x + 2 y = 12$ $3 (- 2) + 2 y = 12$ $- 6 + 2 y = 12$ $2 y = 18$ $y = 9$

$(- 2, 9)$ is a solution.

Plot $(4, 0)$ , $(0, 6)$ , and $(- 2, 9)$ .

Draw the line through the three points.

Find the x-intercept and the y-intercept for the graph of $2 x - y = 8$ . Then draw the graph of $2 x - y = 8$ .

Find the x-intercept by substituting 0 for y in $2 x - y = 8$ and then solving for x.

$2 x - 0 = 8$ $2 x = 8$ $x = 4$

The x-intercept is $(4, 0)$ .

Find the y-intercept by substituting 0 for x in $2 x - y = 8$ and then solving for y.

$2 (0) - y = 8$ $0 - y = 8$ $- y = 8$ $y = - 8$

The y-intercept is $(0, - 8)$ .

Sometimes using the intercepts to draw the graph is awkward for practical reasons.

Find the x-intercept and the y-intercept for the graph of $- 7 x + 2 y = 42$ . Then draw the graph of $- 7 x + 2 y = 42$ .

Find the x-intercept by substituting 0 for y in $- 7 x + 2 y = 42$ and then solving for x.

$- 7 x + 2 (0) = 42$ $- 7 x = 42$ $x = - 6$

The x-intercept is $(- 6, 0)$ .

Find the y-intercept by substituting 0 for x in $- 7 x + 2 y = 42$ and then solving for y.

$- 7 (0) + 2 y = 42$ $2 y = 42$ $y = 21$

The y-intercept is $(0, 21)$ .

Since $(0, 21)$ is not on the Cartesian coordinate system on which I plan to graph, I find another solution.

Substitute $- 4$ for x in $- 7 x + 2 y = 42$ , and then solve for y.

$- 7 (- 4) + 2 y = 42$ $28 + 2 y = 42$ $2 y = 14$ $y = 7$

The point $(- 4, 7)$ is on the graph.

Plot $(- 6, 0)$ and $(- 4, 7)$ . Then draw the line through them.

Draw the graph of $y = - \frac{2}{3} x + 5$ .

Find solutions to $y = - \frac{2}{3} x + 5$ . Since the equation is solved for y, it is easiest to substitute numbers for x and then to find the corresponding values of y.

Substitute 0 for x.

$y = - \frac{2}{3} (0) + 5$ $y = 0 + 5$ $y = 5$

The point $(0, 5)$ is on the graph.

Substitute 2 for x.

$y = - \frac{2}{3} (2) + 5$ $y = - \frac{4}{3} + 5$ $y = - \frac{4}{3} + \frac{15}{3}$ $y = \frac{11}{3}$

The point $(2, \frac{11}{3})$ is on the graph.

Plotting a point with a fractional coordinate is a nuisance. The next number substituted for x will be a multiple of 3 so that its product with $\frac{2}{3}$ will be an integer.

Substitute 6 for x.

$y = - \frac{2}{3} (6) + 5$ $y = - 4 + 5$ $y = 1$

The point $(6, 1)$ is on the graph.

Plot $(0, 5)$ and $(6, 1)$ . Then draw the line through the two points.

Graph $2 x + 5 y = 0$ .

Find the x-intercept by substituting 0 for y and then solving for x.

$2 x + 5 (0) = 0$ $2 x + 0 = 0$ $2 x = 0$ $x = 0$

The x-intercept is $(0, 0)$ .

Find the y-intercept by substituting 0 for x and then solving for y.

$2 (0) + 5 y = 0$ $0 + 5 y = 0$ $5 y = 0$ $y = 0$

The y-intercept is $(0, 0)$ .

The x-intercept and the y-intercept are the same point. It takes two different points to determine a line. Find another point on the graph.

Substitute 10 for x, and then solve for y.

$2 (10) + 5 y = 0$ $20 + 5 y = 0$ $5 y = - 20$ $y = - 4$

The point $(10, - 4)$ is on the graph.

Plot $(0, 0)$ and $(10, - 4)$ . Then draw the line through the two points.

An equation in x and y must not have any variable besides x and y, but it is not required to use both x and y.

Graph $x = 2$ .

Every ordered pair whose x-coordinate is 2 is a solution of the equation $x = 2$ .

Plot $(2, - 7)$ and $(2, 5)$ . Then draw the line through the two points.

Graph $y = 6$ .

Every ordered pair whose y-coordinate is 6 is a solution of the equation $y = 6$ .

Plot $(4, 6)$ and $(- 9, 6)$ . Then draw the line through the two points.

Graph $x + 5 = 1$ .

Solve $x + 5 = 1$ for x.

$x + 5 = 1$ $x = - 4$

Every ordered pair whose x-coordinate is $- 4$ is a solution.

Plot some ordered pairs whose x-coordinates are $- 4$ . Draw the line through the points.

The weight y (in ounces) of a snake longer than 6 inches can be approximated by the linear equation $y = 1.5 x + 5$ , where x is the length of the snake in inches.

Use the equation to approximate the weight of a snake whose length is 12 inches.

Substitute 12 for x in $y = 1.5 x + 5$ .

$y = 1.5 (12) + 5$ $y = 18 + 5$ $y = 23$

The weight of a snake whose length is 12 inches is approximately 23 ounces.

Graphing Linear Equations in `x` and `y`

Graph of an Equation in `x` and `y`

Linear Equation in `x` and `y`

Graphing a Linear Equation

`x`-intercept

`y`-intercept

Graphing Linear Equations in x and y

Graph of an Equation in x and y

Linear Equation in x and y