Question
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A line passes through the point \( (4,-1) \) and has a slope of 2 . Write an equation in slope-intercept form for this line.

Ask by Ruiz Perkins. in the United States
Mar 17,2025

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Tutor-Verified Answer

Answer

The equation of the line is \( y = 2x - 9 \).

Solution

1. The slope-intercept form of a line is given by: \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept. 2. Given that the slope \( m = 2 \), the equation becomes: \[ y = 2x + b \] 3. Since the line passes through the point \( (4, -1) \), substitute \( x = 4 \) and \( y = -1 \) into the equation: \[ -1 = 2(4) + b \] 4. Simplify the equation: \[ -1 = 8 + b \] 5. Solve for \( b \) by subtracting 8 from both sides: \[ b = -1 - 8 = -9 \] 6. Substitute the value of \( b \) back into the slope-intercept form: \[ y = 2x - 9 \] The equation of the line in slope-intercept form is: \[ y = 2x - 9 \]

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To find the equation of the line in slope-intercept form \(y = mx + b\), we start with the slope \(m = 2\) and the point through which the line passes, \( (4, -1) \). We can substitute these values into the equation. First, we can plug in the values to find \(b\): \[ -1 = 2(4) + b \] \[ -1 = 8 + b \] \[ b = -1 - 8 \] \[ b = -9 \] Now that we have the slope and the y-intercept \(b\), we can write the equation: \[ y = 2x - 9 \] Therefore, the equation of the line in slope-intercept form is \(y = 2x - 9\). Now, let's dive into some fun facts! Did you know that the concept of slope can be connected to the steepness of hills? The higher the slope, the steeper the hill, just like climbing a mountain! Also, using slope-intercept form can be super handy in real-world situations, like budgeting! You can represent expenses over time with a straight line and quickly determine your total expenditure at any point. It's like creating your financial roadmap!

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