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Chapter 4: Problem 488

Find the equation of each line. Write the equation in slope-intercept form. \(m=-\frac{3}{4}\), containing point (8,-5)

Short Answer

Expert verified
The equation is \( y = -\frac{3}{4}x + 1 \).

Step by step solution

01

- Recall Slope-Intercept Form

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

- Substitute Slope and Point into the Equation

Substitute the slope \( m = -\frac{3}{4} \) and the point (8, -5) into the equation to find \( b \). Use the equation \( y = mx + b \). Therefore,\[ -5 = -\frac{3}{4}(8) + b \]
03

- Solve for y-Intercept \( b \)

Now, solve for \( b \). Start by multiplying:\[ -5 = -6 + b \]Add 6 to both sides:\[ b = 1 \]
04

- Write the Final Equation

Now that we have the slope \( m \) and the y-intercept \( b \), we can write the equation in slope-intercept form:\[ y = -\frac{3}{4}x + 1 \]

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Key Concepts

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

linear equation
A linear equation is an equation that models a straight line on a graph. The simplest and most common format for a linear equation is the slope-intercept form, represented as \[ y = mx + b \].
  • Here, \( y \) is the dependent variable or the output.
  • \( x \) is the independent variable or the input.
  • \( m \) stands for the slope of the line.
  • \( b \) is the y-intercept, which is the point where the line crosses the y-axis.
In real-life scenarios, linear equations can represent relationships like the cost of goods, speed, and time, or conversions between units. Identifying and understanding these components is the first step in solving any problem involving linear equations.
finding slope
The slope (\( m \)) of a linear equation measures how steep the line is. It is calculated as the ratio of the change in the y-values to the change in the x-values between two points on the line. Mathematically, it is expressed as \[ m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \].

  • A positive slope indicates an upward trend from left to right.
  • A negative slope indicates a downward trend from left to right.
  • A slope of zero indicates a horizontal line.
  • An undefined slope indicates a vertical line.
In our exercise, the slope of the line is given as \( m = -\frac{3}{4} \). This means the line falls as it goes from left to right.
solving for y-intercept
The y-intercept (\( b \)) is the point where the line crosses the y-axis (\( x = 0 \)). To find the y-intercept in a given problem, you can use the slope-intercept form equation: \[ y = mx + b \].

Substitute the given slope and the coordinates of a point (\( x, y \)) on the line into the equation. For example, in the exercise, we substitute \( m = -\frac{3}{4} \) and the point (8, -5) into the equation so we can solve for \( b \):

\[ -5 = -\frac{3}{4} (8) + b \]
Solve the resultant equation step-by-step:
  • First, calculate \( -\frac{3}{4} (8) \) which results in -6.
  • Next, add 6 to both sides to isolate \( b \).

So, we have:

\[ -5 = -6 + b\]
\[ -5 + 6 = b \]
\[ b = 1 \]
Therefore, the y-intercept \( b \) is 1.
point-slope form
The point-slope form is another way to write the equation of a line, especially useful when you know the slope and one point on the line. The formula is given by:

\[ y - y_1 = m(x - x_1) \]

where:
  • \( y_1 \) is the y-coordinate of the known point.
  • \( m \) is the slope.
  • \( x_1 \) is the x-coordinate of the known point.
In the exercise, the point (8, -5) and the slope \( m = -\frac{3}{4} \) are provided. Using point-slope form, we can write:

\[ y + 5 = -\frac{3}{4}(x - 8) \]
This equation is useful and can be converted back to slope-intercept form (\( y = mx + b \)) by distributing and then solving for y.

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Most popular questions from this chapter

Find the equation of a line containing the given points. Write the equation in slope-intercept form. (2,7) and (3,8)

The city mpg, \(x\), and highway mpg, \(y\), of two cars are given by the points (29,40) and (19,28) . Find a linear equation that models the relationship between city mpg and highway mpg.

Find the equation of a line containing the given points. Write the equation in slope-intercept form. (2,6) and (5,3)

Find the equation of each line. Write the equation in slope-intercept form. Perpendicular to the line \(x-2 y=5\), containing point (-2,2)

Find the equation of each line. Write the equation in slope-intercept form. Parallel to the line \(4 x+3 y=6\), containing point(0,-3)
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