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Chapter 0: Problem 30

Find an equation of the line: with \(m=-3,\) containing \((5,-2).\)

Short Answer

Expert verified
The equation of the line is \( y = -3x + 13 \).

Step by step solution

01

Understand the slope-intercept form

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

Substitute the given slope

We are given the slope \(m = -3\). Substitute \(-3\) for \(m\) in the equation: \[ y = -3x + b \]
03

Use the point to find the y-intercept

We are given a point \((5, -2)\) that lies on the line. Substitute \(x = 5\) and \(y = -2\) into the equation to find \(b\): \[-2 = -3(5) + b\]Solve for \(b\):\[-2 = -15 + b \] \[b = 13\]
04

Write the final equation

Now that we have \(b = 13\), substitute it back into the equation: \[ y = -3x + 13 \]

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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 of a linear equation is a very user-friendly way to write the equation of a line. It is written as:
\[ y = mx + b \]
Here, m represents the slope of the line, and b represents the y-intercept, or the point where the line crosses the y-axis.
This form is especially convenient because it clearly shows both the slope and where the line starts on the y-axis.
By looking at the equation, you can quickly determine the steepness and direction of the line as well as its starting point.
finding slope
The slope of a line is a measure of its steepness. It indicates how much the y-value changes for every unit change in the x-value. The slope can be found using two points on the line with the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
In our original exercise, the slope, m, was given as -3. This negative value tells us that the line slopes downward from left to right.
If you are not given the slope and need to find it, simply plug the coordinates of two points on the line into the formula.
y-intercept
The y-intercept is where the line crosses the y-axis. This happens when the value of x is zero.
To find the y-intercept, you can rearrange the equation:
\[ y = mx + b \]
and set x to 0. Solving for y then gives you the y-intercept, b.
In our exercise, we substituted the slope -3 and the coordinates of the point (5, -2) into the slope-intercept form to find our y-intercept:
\[ -2 = -3(5) + b \]
Solving this gave us the y-intercept (b) of 13.
point-slope relationship
The point-slope form of a line's equation is useful when you have one point on the line and the slope. It is written as:
\[ y - y_1 = m(x - x_1) \]
Here, (x_1, y_1) is a point on the line, and m is the slope. For our exercise, we had the point (5, -2) and the slope -3.
Plug these values into the formula:
\[ y - (-2) = -3(x - 5) \]
This becomes:
\[ y + 2 = -3x + 15 \]
To convert this into the slope-intercept form, simplify:
\[ y = -3x + 13 \]
This shows how the point-slope form can be used to find the slope-intercept form.

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