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Chapter 2: Problem 17

Write an equation in slope-intercept form for the line that satisfies each set of conditions. passes through \((-2,5)\) and \((3,1)\)

Short Answer

Expert verified
\( y = -\frac{4}{5}x + \frac{17}{5} \)

Step by step solution

01

Understand the Slope-Intercept Form

The slope-intercept form of a linear equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. To write the equation, we first need to find the slope and then the y-intercept.
02

Calculate the Slope

The slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula \( m = \frac{y_2-y_1}{x_2-x_1} \). Here, point 1 is \((-2,5)\) and point 2 is \((3,1)\). Thus, \( m = \frac{1-5}{3-(-2)} = \frac{-4}{5} = -\frac{4}{5} \).
03

Identify the Slope-Intercept Equation Form

After finding the slope, use point-slope form \( y - y_1 = m(x - x_1) \) to find the equation. Choose one point, say \((-2,5)\), and substitute it along with the slope: \( y - 5 = -\frac{4}{5}(x + 2) \).
04

Simplify to Slope-Intercept Form

Distribute the slope and simplify to find \( b \). \[ y - 5 = -\frac{4}{5}x - \frac{8}{5} \] Add 5 to both sides to solve for \( y \): \[ y = -\frac{4}{5}x - \frac{8}{5} + 5 \] Convert 5 to a fraction with denominator 5: \[ y = -\frac{4}{5}x - \frac{8}{5} + \frac{25}{5} \] Combine the fractions: \[ y = -\frac{4}{5}x + \frac{17}{5} \] Thus, the slope-intercept form is \( y = -\frac{4}{5}x + \frac{17}{5} \).

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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 represent straight lines when plotted on a graph. These equations take the form of \( y = mx + b \), where:
  • \( y \) is the dependent variable (usually the vertical axis).
  • \( x \) is the independent variable (usually the horizontal axis).
  • \( m \) represents the slope of the line.
  • \( b \) is the y-intercept, where the line crosses the y-axis.
Linear equations are fundamental in mathematics because they model relationships where change in one quantity results in a uniform change in another. They are often used to determine trends and make predictions based on observed data. Understanding how to manipulate and work with linear equations is crucial for grasping more advanced mathematical concepts.
Slope Calculation
Calculating the slope of a line involves determining how steep the line is. The slope is a ratio that describes the change in the vertical direction (rise) compared to the horizontal direction (run).To calculate the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\), use the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]This formula finds the difference in the \( y \)-values and the \( x \)-values. It's essential to:
  • Identify the correct points as \((x_1, y_1)\) and \((x_2, y_2)\).
  • Perform the subtraction correctly to avoid confusion.
The slope can be positive, negative, zero, or undefined. A positive slope rises upward from left to right, while a negative slope falls down. A zero slope indicates a horizontal line, and an undefined slope means the line is vertical.
Point-Slope Form
The point-slope form of a linear equation, \( y - y_1 = m(x - x_1) \), is a valuable format for creating the equation of a line when you know a point on the line and the slope.It allows you to plug in:
  • \( (x_1, y_1) \), a known point on the line.
  • \( m \), the slope of the line.
This form is particularly useful because you can immediately write down the equation by substituting values into the formula. It serves as an intermediary step before converting to the slope-intercept form. By manipulating the point-slope form, you can easily solve for \( y \) to find the more commonly used slope-intercept form \( y = mx + b \). This conversion provides a clearer understanding of the line's behavior and overall steepness.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), the \( b \) represents the y-intercept. It is crucial because it provides a starting point for graphing a linear equation.To find the y-intercept from the point-slope form:
  • Start by expanding the equation and solving for \( y \).
  • Move from \( y - y_1 = m(x - x_1) \) to an equation which isolates \( y \).
  • Plug in the slope \( m \) and your known point, and simplify accordingly.
Understanding the y-intercept helps us know where the line hits the y-axis, and it allows for graph plotting and analyzing trends without drawing the entire graph. It simplifies interpreting and making predictions from linear data.

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Most popular questions from this chapter

Graph each inequality. $$ x-5 \leq y $$

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COLLEGE For Exercises 22 and \(23,\) use the following information. Rosa's professor says that the midterm exam will count for 40\(\%\) of each student's grade and the final exam will count for 60\(\% .\) A score of at least 90 is required for an A. Refer to the graph. If she scores 85 on the midterm and 95 on the final, will Rosa get an \(\mathrm{A}\) ?
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