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Chapter 1: Problem 22

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

- Substitute the Slope

We know the slope \( m \) is -3. Substitute \( m = -3 \) into the slope-intercept form equation: \[ y = -3x + b \]
03

- Use the Given Point to Find the Y-Intercept

The line passes through the point (5, -2). Substitute \( x = 5 \) and \( y = -2 \) into the equation you've found: \[ -2 = -3(5) + b \] Simplify to solve for \( b \).
04

- Solve for the Y-Intercept

Solving the equation from the previous step:\[ -2 = -15 + b \] Add 15 to both sides to isolate \( b \): \[ b = 13 \]
05

- Write the Final Equation

Substitute \( b = 13 \) back into the equation from Step 2: \[ y = -3x + 13 \] This is the equation of the line.

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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 way of writing the equation of a line so that you can easily identify the slope and the y-intercept. The general formula for this form is \[ y = mx + b \] where:
  • \( y \) represents the y-coordinate of any point on the line.
  • \( x \) represents the x-coordinate of any point on the line.
  • \( m \) is the slope of the line.
  • \( b \) is the y-intercept of the line.
The slope-intercept form is very useful because it immediately tells you both the slope (how steep the line is) and the y-intercept (where the line crosses the y-axis). This makes it easy to graph the line or find specific points.
finding the slope
The slope of a line is a measure of how steep it is. It is represented by the letter \( m \) in the slope-intercept form equation. The slope can be calculated if you know two points on the line. The formula for finding the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]In our exercise, we are given that the slope \( m \) is -3. This means that for every unit increase in \( x \), the value of \( y \) decreases by 3 units. Knowing the slope is essential for writing the equation in slope-intercept form.
finding the y-intercept
The y-intercept is the point where the line crosses the y-axis. In the slope-intercept form equation \( y = mx + b \), the intercept is represented by \( b \). To find it, you can use a known point on the line and the slope. Given the point \((5, -2)\) from the exercise, substitute \( x \) and \( y \) into the slope-intercept form:\[ -2 = -3(5) + b \]Simplify the equation to find \( b \): \[ -2 = -15 + b \] Add 15 to both sides: \[ b = 13 \]Thus, the y-intercept is 13.
linear equations
A linear equation is an equation that makes a straight line when it is graphed. Linear equations have no exponents higher than 1. That's why they are called 'linear.' The standard forms of linear equations are:
  • Slope-intercept form: \( y = mx + b \)
  • Standard form: \( Ax + By = C \)
In the exercise, we focused on the slope-intercept form to find the equation of a line with a provided slope and a point. Knowing these forms can help you convert between them and understand the relationship between the algebraic equation and the geometric line. Finally, the equation we derive in slope-intercept form is: \[ y = -3x + 13 \]

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