Question

write an equation (a) in slope - intercept form and (b) in standard form for the line passing through (-1, 4) and parallel to x + 4y = 7.

Explanation:

Step1: Find the slope of the given line

First, rewrite the equation \(x + 4y = 7\) in slope - intercept form \(y=mx + b\) (where \(m\) is the slope and \(b\) is the y - intercept).
Solve for \(y\):
\(4y=-x + 7\)
\(y=-\frac{1}{4}x+\frac{7}{4}\)
The slope of the line \(x + 4y = 7\) is \(m =-\frac{1}{4}\). Since parallel lines have the same slope, the line we want to find also has a slope \(m=-\frac{1}{4}\).

Step2: Use the point - slope form to find the equation of the line

The point - slope form of a line is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)\) is a point on the line and \(m\) is the slope. We know that \(m =-\frac{1}{4}\) and the point \((x_1,y_1)=(-1,4)\).
Substitute these values into the point - slope form:
\(y - 4=-\frac{1}{4}(x+ 1)\)

Step3: Rewrite in slope - intercept form (part a)

Expand the right - hand side:
\(y - 4=-\frac{1}{4}x-\frac{1}{4}\)
Add 4 to both sides. To add 4 and \(-\frac{1}{4}\), we write 4 as \(\frac{16}{4}\). So:
\(y=-\frac{1}{4}x-\frac{1}{4}+\frac{16}{4}\)
\(y=-\frac{1}{4}x+\frac{15}{4}\)

Step4: Rewrite in standard form (part b)

The standard form of a line is \(Ax+By = C\), where \(A\), \(B\), and \(C\) are integers and \(A\geq0\).
Start with the slope - intercept form \(y=-\frac{1}{4}x+\frac{15}{4}\).
Multiply every term by 4 to eliminate the fractions:
\(4y=-x + 15\)
Add \(x\) to both sides:
\(x + 4y=15\)

Answer:

(a) The equation in slope - intercept form is \(y =-\frac{1}{4}x+\frac{15}{4}\).
(b) The equation in standard form is \(x + 4y=15\).