Question

What is the slope, x-intercept and y-intercept of the graph of $3x+y=7$?

Answer 1

Hint: Here, we can directly rearrange the given equation to convert it into the slope intercept form. Thus, we will get the value of the slope. We can then again convert the given equation into the intercept form, and then we can easily calculate the x-intercept and the y-intercept.

Complete step-by-step solution:
We know that the slope intercept form of the equation of a straight line is $y=mx+c$, where m is the slope of this straight line.
Here, in this question, we are given the equation of line $3x+y=7$.
Let us rewrite this equation in the slope intercept form. We get
$y=-3x+7...\left( i \right)$
On comparing equation (i) with the general slope intercept form, we get
$m=-3$
Hence, the slope of this equation is -3.

We also know that the intercept form of a straight line is $\dfrac{x}{a}+\dfrac{y}{b}=1$, where a is the intercept on x-axis, and b is the intercept on the y-axis.
We are given the equation $3x+y=7$.
Let us divide both the left hand side and the right hand side of this equation by 7. Hence, we get
$\dfrac{3x+y}{7}=\dfrac{7}{7}$
We can simplify the above equation to get
$\dfrac{3x}{7}+\dfrac{y}{7}=1$
To change this equation into the intercept form of a straight line, we need to have the coefficients of x and y as 1 in the numerators.
So, rewriting out evaluated equation, we get
$\dfrac{x}{\left( \dfrac{7}{3} \right)}+\dfrac{y}{7}=1...\left( ii \right)$
We now have equation (ii) in the intercept form and we can make a comparison with the general form of the intercept form of the equation of a straight line. So, on comparing, we get
$a=\dfrac{7}{3}$ and $b=7$.
So, the length of x-intercept = $\dfrac{7}{3}$
and, the length of y-intercept = $7$.
Hence, slope is $-3$, x-intercept is $\dfrac{7}{3}$ and y-intercept is $7$.

Note: We must take care that the denominator of x is the x-intercept, and the denominator of y is the y-intercept. Also, it is worth mentioning, that the right hand side part of an equation in intercept form is always equal to 1. We can also use the values of a and b to calculate the slope.