How to Write the Slope-Intercept Form of the Equation of a Line Parallel to 3x 4y -32 and Passing Through -4, 1
Have you ever tried writing the equation of a line that is parallel to another and also passes through a specific point? This guide will walk you through the steps to find the slope-intercept form of the equation of a line that passes through the point -4, 1 and is parallel to the line given by the equation 3x 4y -32.
Step 1: Find the Slope of the Given Line
To begin with, we need to rewrite the given equation in slope-intercept form, which is expressed as y mx b where m is the slope and b is the y-intercept.
The given equation is:
[ 3x 4y -32 ]Step 1.1: Isolate y on one side of the equation.
[ 4y -3x - 32 ]Step 1.2: Divide every term by 4 to solve for y.
[ y -frac{3}{4}x - 8 ]From this equation, we can see that the slope m of the given line is -frac{3}{4}. Since parallel lines have the same slope, the slope of our desired line is also -frac{3}{4}.
Step 2: Use the Point-Slope Form
The point-slope form of the equation of a line is given by:
[ y - y_1 m(x - x_1) ]where (x_1, y_1) is a point on the line. In our case, the point is -4, 1.
Step 3: Substitute the Values
Substitute the point (-4, 1) and the slope -frac{3}{4} into the point-slope form.
[ y - 1 -frac{3}{4}(x 4) ]Step 4: Simplify to Slope-Intercept Form
Step 4.1: Distribute the slope.
[ y - 1 -frac{3}{4}x - 3 ]Step 4.2: Add 1 to both sides to solve for y.
[ y -frac{3}{4}x - 2 ]The final equation of the line in slope-intercept form is:
[ y -frac{3}{4}x - 2 ]Alternative Method Using the Given Line’s Equation
Another method involves directly finding the equation in the form 3x 4y c. Since it passes through the point (-4, 1), substitute x -4 and y 1 into the equation:
[ 3(-4) 4(1) c ]This simplifies to:
[ -12 4 c ]Thus, c -8, and the equation of the line is:
[ 3x 4y -8 ]Converting this to slope-intercept form, we get:
[ y -frac{3}{4}x - 2 ]As you can see, both methods lead to the same slope-intercept form.
Conclusion
The slope-intercept form of the equation of the line that passes through the point -4, 1 and is parallel to the line 3x 4y -32 is:
[ boxed{y -frac{3}{4}x - 2} ]Understanding these steps can help you tackle similar problems in the future, ensuring you can efficiently find the equations of parallel lines passing through specific points.