Exercise 31 Page 495

We will solve the equation by graphing each side of the equation. The

x -

coordinates of the points of intersection of the graphs are the solutions of the equation.

5 x - 7 = x^{2} \Rightarrow LHS : y = 5 x - 7 RHS : y = x^{2}

Let's graph the functions related to the left-hand side and the right-hand side of the equation one at a time.

Left-Hand Side

We will start by graphing the linear function

y = 5 x - 7 .

Let's rewrite the equation of the first function in slope-intercept form. Also we can highlight the slope

m

and the

y -

intercept

b .

y = 5 x - 7 \Leftrightarrow y = 5 x + (- 7)

Plot the

y -

intercept and use the slope to plot another point. Then we will connect the points with a straight edge.

Right-Hand Side

Now we will graph the function $y = x^{2} .$ It is the parent quadratic function. The vertex of the graph of $y = x^{2}$ is $(0, 0)$ and its axis of symmetry is $x = 0,$ or the $y -$ axis. Next, let's make a table of values to help us with graphing.

$x$	$y = x^{2}$
$1$	$1$
$2$	$4$
$3$	$9$

Plot these point and their reflection across the axis of symmetry.

Finally, we will connect the points with a smooth curve.

Connecting the Graphs

The last step is connecting the graphs and determining their points of intersection. Recall that the $x -$ coordinates of the points of intersection are the solutions of the given equation.

The graphs do not intersect. Therefore, the equation $5 x - 7 = x^{2}$ has no real solutions.