Question

How do you find the x and y-intercept of $4x - 7y = 84$?

Answer 1

This is a linear equation of the first order. In this question, a linear equation is given. We will convert this equation into the form of a straight-line equation. Then the value of x is equal to 0 to find the y-intercept. Then convert this equation in terms of x. And put the value of y is equal to 0 to find the x-intercept.

Complete step by step solution:
In this question, the linear equation is
$\Rightarrow 4x - 7y = 84$
Let us subtract 4x on both sides.
$\Rightarrow 4x - 4x - 7y = 84 - 4x$
Therefore,
$\Rightarrow - 7y = 84 - 4x$
Now, let us divide both sides by -7.
$\Rightarrow y = \dfrac{{84 - 4x}}{{ - 7}}$
Let us split the denominator.
$\Rightarrow y = \dfrac{{84}}{{ - 7}} - \dfrac{{4x}}{{ - 7}}$
Now, simplify the right-hand side.
$\Rightarrow y = - 12 + \dfrac{{4x}}{7}$
That is equal to,
$\Rightarrow y = \dfrac{4}{7}x - 12$ ...(1)
Now, to find the value of the y-intercept we will put the value of x is 0 in equation (1).
So, the y-intercept is,
$\Rightarrow y = - 12$
Now, to find the value of the x-intercept we will put the value of y as 0 in equation (1).
$\Rightarrow 0 = \dfrac{4}{7}x - 12$
Let us add 12 on both sides.
$\Rightarrow 0 + 12 = \dfrac{4}{7}x - 12 + 12$
Now, find the least common multiple of the denominator on the right-hand side.
$\Rightarrow 12 = \dfrac{4}{7}x$
Let us multiply both sides by $\dfrac{7}{4}$ .
$\Rightarrow 12 \times \dfrac{7}{4} = \dfrac{4}{7}x \times \dfrac{7}{4}$
That is equal to
$\Rightarrow x = 21$

Hence, the value of the x-intercept is 21 and the value of the y-intercept is -12.

Note:
The x-intercept is where a line crosses the x-axis, and the y-intercept is the point where the line crosses the y-axis. In the question, we can say that the value of the x-intercept is 21. So, the point on the x-axis is $\left( {21,0} \right)$. And the value of the y-intercept is -12. So, the point on the y-axis is $\left( {0, - 12} \right)$.