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Chapter 1: Problem 26

You have been using the form \(y=m x+b\) to represent linear equations. Linear equations are sometimes represented in the form \(A x+B y=C,\) where \(A, B,\) and \(C\) are constants. a. Rewrite the equation \(A x+B y=C\) in the \(y=m x+b\) form. To do this, you will need to express \(m\) and \(b\) in terms of \(A, B,\) and \(C .\) b. What is the slope of a line with an equation in the form \(A x+B y=C ?\) What is the \(y\) -intercept?

Short Answer

Expert verified
Slope: \( -\frac{A}{B} \), y-intercept: \( \frac{C}{B} \).

Step by step solution

01

Identify the Given Equation

Given the equation in the form: \[ A x + B y = C \].
02

Solve for y

To rewrite the equation in the form \( y = m x + b \), solve for \( y \) in terms of \( x \): Start by isolating the \( y \)-term: \[ B y = -A x + C \]. Next, solve for \( y \) by dividing every term by \( B \): \[ y = -\frac{A}{B} x + \frac{C}{B} \].
03

Identify the Slope and y-Intercept

From the equation \( y = -\frac{A}{B} x + \frac{C}{B} \), compare it with the standard form \( y = m x + b \): The slope \( m \) is \( -\frac{A}{B} \). The y-intercept \( b \) is \( \frac{C}{B} \).

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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 linear equation format written as \( y = mx + b \).
This form is very useful because it easily shows the slope and y-intercept of a line.
Literally, it breaks down into two parts:
  • \( m \): The slope of the line, which shows how steep the line is.
  • \( b \): The y-intercept, the point where the line crosses the y-axis.
For example, in the equation \( y = 2x + 3 \), the slope is 2, and the y-intercept is 3.
This means the line rises by 2 units for every 1 unit it moves to the right and crosses the y-axis at \( y = 3 \).
standard form
The standard form of a linear equation is written as \( Ax + By = C \), where \( A, B, \) and \( C \) are constants.
It is useful in situations where you want to easily determine whether a particular point is on the line or for finding intercepts quickly.
It isn't as straightforward as the slope-intercept form for finding the slope and y-intercept directly.
Let's rewrite it into slope-intercept form for clarity:
  • Start with the standard form: \( Ax + By = C \).
  • Solve for \( y \): \( By = -Ax + C \).
  • Divide by \( B \): \( y = -\frac{A}{B}x + \frac{C}{B} \).
Now, we see it in the \( y = mx + b \) form, making it easy to identify the slope and y-intercept.
slope and y-intercept
Understanding the slope and y-intercept is crucial for graphing linear equations and interpreting linear relationships.
#### Slope:The slope, \( m \), indicates the steepness of the line and its direction.
It is calculated as the ratio of the change in \( y \) (rise) to the change in \( x \) (run).
If \( m \) is positive, the line rises; if it is negative, the line falls.
In the converted form \( y = -\frac{A}{B}x + \frac{C}{B} \), the slope is \( -\frac{A}{B} \).#### Y-intercept:The y-intercept, \( b \), is the point where the line crosses the y-axis (when \( x = 0 \)).
It provides a starting point for plotting the graph.
In our example, from \( y = -\frac{A}{B}x + \frac{C}{B} \), the y-intercept is \( \frac{C}{B} \).
Knowing both the slope and y-intercept allows you to sketch the graph of the linear function easily.

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Most popular questions from this chapter

The lines for these three equations all pass through a common point. $$y=\frac{x}{2}-1 \quad y=-\frac{2 x}{3}+6 \quad y=-\frac{x}{6}+3$$ a. Draw graphs for the three equations, and find the common point. b. Verify that the point you found satisfies all three equations by substituting the x- and y-coordinates into each equation.

Alejandro looked at the equations \(y=\frac{3}{2} x-1\) and \(y=-\frac{2}{3} x+2\) and said, These lines form a right angle. a. Graph both lines on two different grids, with the axes labeled as shown here. b. Compare the lines on each grid. Do both pairs of lines form a right angle? c. What kind of assumption must Alejandro have made when he said the lines form a right angle?

Use the distributive property to rewrite each expression without using parentheses. \(p q\left(\frac{1}{p^{2}}-\frac{q}{p}\right)\)

One day Lydia walked from Allentown to Brassville at a constant rate of 4 kilometers per hour. The towns are 30 kilometers apart. a. Write an equation for the relationship between the distance Lydia traveled, \(d,\) and the hours she walked, \(h\). b. Graph your equation to show the relationship between hours walked and distance traveled. Put distance traveled on the vertical axis. c. How many hours did it take Lydia to reach Brassville? d. Now write an equation for the relationship between the hours walked, \(h\), and the distance remaining to complete the trip, \(r\). e. Graph the equation you wrote for Part d on the same set of axes you used for Part b. Label the vertical axis for both \(d\) and \(r\). f. How can you use your graph from Part e to determine how many hours it took Lydia to reach Brassville?

Just as the \(y\) -intercept of a line is the \(y\) value at which the line crosses the \(y\) -axis, the \(x\) -intercept is the \(x\) value at which the line crosses the \(x\) -axis. In Exercises \(22-25,\) find an equation of a line with the given \(x\) -intercept and slope. \(x\) -intercept \(1,\) slope \(-6\)
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