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Chapter 3: Problem 31

Find the slope and the \(y\) -intercept of each line whose equation is given. $$2 x-5 y=-8$$

Short Answer

Expert verified
The slope is \(\frac{2}{5}\) and the \(y\)-intercept is \(\frac{8}{5}\).

Step by step solution

01

Rewrite the equation in 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. First, solve the given equation for \(y\).Starting with the equation: \[2x - 5y = -8\]Isolate \(-5y\) by subtracting \(2x\) from both sides:\[-5y = -2x - 8\]
02

Solve for \(y\)

Next, divide each term on both sides by \(-5\) to solve for \(y\):\[y = \frac{-2x - 8}{-5}\]Simplify the expression:\[y = \frac{2}{5}x + \frac{8}{5}\]Now the equation is in the form \(y = mx + b\).
03

Identify the slope and the \(y\)-intercept

In the equation \(y = \frac{2}{5}x + \frac{8}{5}\), the coefficient of \(x\) is the slope \(m\) and the constant term is the \(y\)-intercept \(b\).So, the slope \(m\) is \(\frac{2}{5}\) and the \(y\)-intercept \(b\) is \(\frac{8}{5}\).

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Key Concepts

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

Understanding Linear Equations
Linear equations are fundamental in algebra. They represent lines on the coordinate plane. The general form of a linear equation is written as: \[ax + by = c\] Here,
  • a and b are coefficients
  • x and y are variables
  • c is a constant
Solving linear equations involves finding the values of x and y that satisfy the equation. A common method of analyzing a linear equation is to convert it into a form that easily reveals its characteristics, such as the slope-intercept form.
The Slope-Intercept Form of a Linear Equation
The slope-intercept form is a way to express linear equations for easy analysis. It is written as: \[y = mx + b\] Here:
  • m represents the slope of the line
  • b represents the y-intercept, the point where the line crosses the y-axis
Converting an equation to this form helps in directly identifying the slope and y-intercept. For example, let's consider the provided equation: \[2x - 5y = -8\] To convert it to slope-intercept form, follow these steps:
  • Subtract 2x from both sides: \[-5y = -2x - 8\]
  • Divide all terms by -5: \[y = \frac{2}{5}x + \frac{8}{5}\]
Now the equation is in the form \[y = mx + b\], revealing the slope (`\frac{2}{5}`) and the y-intercept (`\frac{8}{5}`).
Solving for y
To solve for y means to isolate y on one side of the equation. This process is crucial when converting an equation to the slope-intercept form. Let's revisit the original problem step-by-step:
  • Start with the given equation: \[2x - 5y = -8\]
  • Subtract 2x from both sides to isolate the term with y: \[-5y = -2x - 8\]
  • Divide each term by -5 to solve for y: \[y = \frac{-2x - 8}{-5}\] Notice, signs change when dividing by a negative number: \[y = \frac{2}{5}x + \frac{8}{5}\]
Now, y is isolated, and the equation is in slope-intercept form, making it easy to identify the slope and y-intercept. This approach can be used for any linear equation to reveal its crucial characteristics.

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