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Chapter 2: Problem 16

Write an equation in slope-intercept form for the line that satisfies each set of conditions. slope \(\frac{3}{2}\) passes through \((-5,1)\)

Short Answer

Expert verified
The equation is \( y = \frac{3}{2}x + \frac{17}{2} \).

Step by step solution

01

Understand Slope-Intercept Form

The slope-intercept form of a linear equation is \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.
02

Identify Given Values

The problem states that the slope is \( \frac{3}{2} \) and the line passes through the point \((-5, 1)\). This means \( m = \frac{3}{2} \), \( x = -5 \), and \( y = 1 \). We need to find \( b \).
03

Substitute Values into Slope-Intercept Equation

Substitute the given slope \( m = \frac{3}{2} \) and the point \((-5, 1)\) into the equation \( y = mx + b \). This gives us \( 1 = \frac{3}{2}(-5) + b \).
04

Solve for the Y-Intercept \( b \)

Calculate \( \frac{3}{2} \times -5 = -\frac{15}{2} \). Then, substitute this back into the equation: \( 1 = -\frac{15}{2} + b \). To find \( b \), add \( \frac{15}{2} \) to both sides:\[ b = 1 + \frac{15}{2} = \frac{2}{2} + \frac{15}{2} = \frac{17}{2} \]
05

Write the Final Equation

Now that we have both \( m \) and \( b \), we can write the equation of the line in slope-intercept form as: \( y = \frac{3}{2}x + \frac{17}{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
The slope-intercept form is a useful way to write the equation of a line in algebra. This form is represented by the formula \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) represents the y-intercept.
  • **Slope** \( (m) \): It shows the rise over run, indicating how steep the line is. A positive slope means the line ascends, while a negative slope means the line descends.
  • **Y-Intercept** \( (b) \): This is the point where the line crosses the y-axis. It is key in determining the exact position of the line on a graph.
Slope-intercept form is particularly valuable because it provides an immediate understanding of the line's direction and position through both its slope and y-intercept.
This form is often used because it's straightforward, allows easy graphing, and enables simple calculations when altering or analyzing a line.
Point-Slope Form
Sometimes, you won't have the y-intercept given. Instead, you might have a point on the line and the slope鈥攖hat's where point-slope form comes in handy. The formula is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is a specific point on the line.
  • It rearranges easily to find the equation in slope-intercept form.
  • Helps quickly determine the equation of a line if only the slope and a single point are known.
An example can illustrate its use: Given the point \((-5, 1)\) and a slope of \(\frac{3}{2}\), you can plug these into the formula:
\( y - 1 = \frac{3}{2}(x + 5) \). Calculating this gives you the line's equation, which you can transform into slope-intercept form.
Finding the Y-Intercept
To find the y-intercept, you must determine where a line will cross the y-axis. This point is crucial when writing the equation in slope-intercept form.
Using given points or the slope and a point, you can substitute into the equation \( y = mx + b \) to solve for \( b \).
  • Start with substituting the slope \( m \) and the coordinates of the known point \( (x, y) \) into the equation.
  • Solve for \( b \) by rearranging and isolating the variable.
For instance, given the point \((-5, 1)\) and slope \(\frac{3}{2}\), substitute these values into the equation:
\( 1 = \frac{3}{2}(-5) + b \). Solve for \( b \) by adding \( \frac{15}{2} \) to both sides, resulting in \( b = \frac{17}{2} \). This process easily finds the y-intercept and completes the linear equation in slope-intercept form.

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Most popular questions from this chapter

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