Question

Find an equation of the line satisfying the conditions. Write your answer in the slope-intercept form. Passes through \((4,-3)\) with slope 2

Short answer

The equation of the line that passes through the point \((4, -3)\) with a slope of \(2\) is \(y = 2x - 11\).

Step-by-step solution

Use slope-intercept form and given information

The general equation of a line is given by \(y = mx + b\). We need to determine the y-intercept \(b\) to complete the equation. As we are given the slope \(m\) (i.e., \(m = 2\)) and a point \(P(4, -3)\) through which the line passes, substitute the values of the coordinates of \(P\) and the slope in the equation to solve for \(b\): \[-3 = 2(4) + b\]

Solve for b (y-intercept)

Solve for \(b\) in the equation \(-3 = 2(4) + b\): \begin{align*} -3 &= 2(4) + b \\ -3 &= 8 + b \\ b &= -11 \end{align*} So, the y-intercept is \(b = -11\).

Write the final equation

Now that we have the slope \(m = 2\) and y-intercept \(b = -11\), we can write the equation of the line in slope-intercept form as follows: \[y = 2x - 11\] So, the equation of the line is: \(y = 2x - 11\).