A line passes through the point $(-8,5)$ and has a slope of $\frac{3}{4}$. Write an equation in slope-intercept form for this line.
Answer (1)
Start with the point-slope form: y − y 1 = m ( x − x 1 ) .
Substitute the given point and slope: y − 5 = 4 3 ( x + 8 ) .
Distribute and simplify: y − 5 = 4 3 x + 6 .
Convert to slope-intercept form: y = 4 3 x + 11 .
y = 4 3 x + 11
Explanation
Understanding the Problem We are given a point ( − 8 , 5 ) and a slope m = 4 3 . We want to find the equation of the line in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept.
Using Point-Slope Form We can use the point-slope form of a line, which is y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is the given point and m is the slope.
Substituting Values Substitute the given point ( − 8 , 5 ) and slope 4 3 into the point-slope form: y − 5 = 4 3 ( x − ( − 8 )) .
Simplifying Simplify the equation: y − 5 = 4 3 ( x + 8 ) .
Distributing Distribute the 4 3 : y − 5 = 4 3 x + 4 3 ( 8 ) .
Simplifying Further Simplify further: y − 5 = 4 3 x + 6 .
Isolating y Isolate y to get the slope-intercept form: y = 4 3 x + 6 + 5 .
Final Equation Write the final equation in slope-intercept form: y = 4 3 x + 11 .
Examples
Understanding linear equations is crucial in many real-world scenarios. For instance, if you're tracking the cost of a taxi ride, the initial fee is the y-intercept, and the cost per mile is the slope. By knowing these values, you can predict the total cost of any ride. Similarly, in business, linear equations can model revenue or expenses, helping to forecast profits or losses based on production levels or sales volumes. This understanding provides a practical tool for making informed decisions in everyday situations.
