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Chapter 3: Problem 22

Write the equation in slope-intercept form of the line satisfying the given conditions. Slope \(-\frac{3}{4} ; y\) -intercept \((0,7)\)

Short Answer

Expert verified
\( y = -\frac{3}{4}x + 7 \)

Step by step solution

01

Understand Slope-Intercept Form

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

Identify the Given Values

From the given conditions, the slope \( m = -\frac{3}{4} \) and the y-intercept \( b = 7 \).
03

Substitute the Values into the Equation

Substitute \( m = -\frac{3}{4} \) and \( b = 7 \) into the slope-intercept form: \[ y = -\frac{3}{4}x + 7 \]

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Key Concepts

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

linear equations
Linear equations are mathematical expressions that describe a straight line on a coordinate plane. They incorporate variables, constants, and coefficients to create a relationship between the x and y values.

In the equation of a line, y is typically the dependent variable, meaning it depends on the value of x, the independent variable. The general form of a linear equation is:

\[ ax + by + c = 0 \]

where:
  • a is the coefficient of x
  • b is the coefficient of y
  • c is the constant term

However, the most common form used for graphing is the slope-intercept form, which simplifies the understanding of the line's properties.
slope
The slope of a line measures its steepness and direction. It is represented by the letter m in the slope-intercept form of a linear equation. Mathematically, the slope is defined as the ratio of the change in the y value to the change in the x value between two points.

The formula for slope is:

\[ m = \frac{\Delta y}{\Delta x} \]

or

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

A positive slope means the line ascends from left to right, while a negative slope means it descends. In our example, the slope is \( -\frac{3}{4} \) indicating a gradual decline, since the change in y is negative when x increases.
y-intercept
The y-intercept is the point at which the line crosses the y-axis. It is denoted by the letter b in the slope-intercept form of a linear equation. This value tells you where the line starts on the y-axis when x is zero.

In practical terms, if you imagine a graph, the y-intercept is the 'starting height' of the line. It gives you a fixed point to plot before using the slope to determine the line鈥檚 direction and steepness.

For our given example:
  • y-intercept: (0, 7)
This means that when x is 0, y is 7. This point (0,7) is crucial information for graphing or understanding the equation since it provides a concrete reference point for constructing the line.

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Most popular questions from this chapter

Find the midpoint of each segment with the given endpoints. $$ (2.5,3.1) \text { and }(1.7,-1.3) $$

Three points that lie on the same straight line are said to be collinear. Consider the points \(A(3,1), B(6,2),\) and \(C(9,3) .\) Find the slope of segment \(A B\)

Write each inequality or compound inequality using interval notation. See Sections 1.1 and 2.6. $$ -4 \leq x \leq 4 $$

Find an equation of the line that satisfies the given conditions. (a) Write the equation in slope-intercept form. (b) Write the equation in standard form. Through \((-1,3) ;\) parallel to \(-x+3 y=12\)

Find the midpoint of each segment with the given endpoints. $$ (5,2) \text { and }(-1,8) $$
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