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Chapter 3: Problem 14

Write an equation of the line passing through the given point and having the given slope. Give the final answer in slope-intercept form. \((-5,4), m=-1\)

Short Answer

Expert verified
The equation of the line is \( y = -x - 1 \).

Step by step solution

01

- Understand the 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

- Substitute the Given Slope

Substitute the given slope \( m = -1 \) into the slope-intercept form equation: \[ y = -x + b \]
03

- Plug in the Given Point

Use the given point \((-5, 4)\) to find the y-intercept \( b \). Substitute \( x = -5 \) and \( y = 4 \) into the equation: \[ 4 = -(-5) + b \]
04

- Solve for the Y-Intercept

Solve the equation to find \( b \): \[ 4 = 5 + b \] Subtract 5 from both sides: \[ b = -1 \]
05

- Write the Final Equation

Now that both the slope \( m \) and y-intercept \( b \) are known, write the final equation: \[ y = -x - 1 \]

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Key Concepts

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

linear equations
Linear equations describe a straight line on a coordinate plane. They take various forms, but the most common is the slope-intercept form. This form is written as:
y = m x + b
where:
  • y is the dependent variable

  • x is the independent variable

  • m is the slope of the line

  • b is the y-intercept


The slope defines how steep the line is, and the y-intercept is the point where the line crosses the y-axis. A slope can be positive, negative, zero, or undefined. Understanding how to construct linear equations helps in graphing lines and solving many algebraic problems.
solving for y-intercept
To find the y-intercept of a linear equation, we need to substitute a known point on the line into the equation and solve for b. Let's break down the steps with a practical example.

Given the point (-5, 4) and the slope m = -1, we start by substituting the slope into the slope-intercept form equation: y = -x + b. Now, using the given point, we substitute x = -5 and y = 4 into the equation: 4 = -(-5) + b. This simplifies to: 4 = 5 + b. To isolate b, subtract 5 from both sides of the equation: b = -1.

This means the line crosses the y-axis at -1. The y-intercept provides a crucial part of the full equation of the line.
substitute point in equation
One effective method to find unknown variables in linear equations is by substituting a given point into the equation. This involves the following steps:
  • Identify the point that lies on the line. In this case, it's (-5, 4).

  • Insert the x and y values from this point into the equation where y and x are.


For instance, if you're working with the equation y = - x + b, substitute x = -5 and y = 4: 4 = -(-5) + b. Simplify to: 4 = 5 + b. Finally, solve for b to find the specific y-intercept. Substituting points into equations helps you determine values for unknown variables, making it easier to build accurate linear equations.

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Most popular questions from this chapter

Write an equation of the line satisfying the given conditions. Give the final answer in slope-intercept form. (Hint: Recall the relationships among slopes of parallel and perpendicular lines.) Passes through (2,3) parallel to \(4 x-y=-2\)

\(2 x-5 y=10\) \(5 x+2 y=12\)

$$ y=\frac{5}{2} x-4 $$

Write an equation of the line satisfying the given conditions. Give the final answer in slope-intercept form. (Hint: Recall the relationships among slopes of parallel and perpendicular lines.) Perpendicular to \(x-2 y=7\) \(y\) -intercept (0,-3)

Write an equation of the line passing through the given pair of points. Give the final answer in (a) slope-intercept form and (b) standard form. (-2,-1) and (3,-4)
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