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Chapter 7: Problem 9

Write an equation in slope-intercept form to represent the table of values. $$\begin{array}{|c|c|c|c|c|} \hline x & -4 & 0 & 4 & 8 \\ \hline y & -4 & -1 & 2 & 5 \\ \hline \end{array}$$

Short Answer

Expert verified
The equation in slope-intercept form is \( y = \frac{3}{4}x - 1 \).

Step by step solution

01

Find the Slope

First, calculate the slope using the formula for slope: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Select two points from the table, for instance, \((x_1, y_1) = (-4, -4)\) and \((x_2, y_2) = (0, -1)\). Substitute into the formula: \( m = \frac{-1 - (-4)}{0 - (-4)} = \frac{3}{4} \). Hence, the slope \( m \) is \( \frac{3}{4} \).
02

Use a Point to Determine y-intercept

With the slope \( m = \frac{3}{4} \) known, use one of the points to find the y-intercept \( b \). Using \((0, -1)\) which is already in y-intercept form, we know directly that \( b = -1 \). However, to confirm, you can substitute \((x, y)\) from the point into the equation \( y = mx + b \) using \((4, 2)\) for a cross-check: \( 2 = \frac{3}{4} \times 4 + b \). Solve for \( b \): \( 2 = 3 + b \), so \( b = -1 \).
03

Write the Slope-Intercept Form Equation

Now that both the slope \( m = \frac{3}{4} \) and the y-intercept \( b = -1 \) are known, write the equation in slope-intercept form: \( y = \frac{3}{4}x - 1 \).

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Key Concepts

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

Slope-Intercept Form
To grasp linear equations better, it's essential to understand the slope-intercept form. This form is written as \( y = mx + b \), where each component has a distinct role:
  • \( y \) represents the dependent variable or the value you solve for.
  • \( m \) is the slope of the line, indicating how steep the line is.
  • \( x \) is the independent variable or the input value.
  • \( b \) indicates the y-intercept, the point where the line crosses the y-axis.
In the slope-intercept form, you can immediately identify both the slope and y-intercept, which makes graphing and understanding linear relationships easier.
Slope Calculation
The slope of a line reflects how steep it is, and it's a crucial element of linear equations. You calculate it using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
For instance, using points \((-4, -4)\) and \((0, -1)\) from the table:
  • Subtract the y-values: \( y_2 - y_1 = -1 - (-4) = 3 \)
  • Subtract the x-values: \( x_2 - x_1 = 0 - (-4) = 4 \)
Putting it together, the slope \( m = \frac{3}{4} \). This tells you that for every increase of 4 units in the x-direction, the y-value increases by 3 units. Remember, a positive slope suggests an upward trend from left to right.
y-intercept Determination
Finding the y-intercept is often straightforward, especially when one of the points lies on the y-axis. In the point \((0, -1)\), the x-value is zero, which means the y-value, \(-1\), is directly the y-intercept \(b\).
If such a point isn't available, you can still determine \(b\) by using the slope-intercept equation \( y = mx + b \) and substituting known values. As demonstrated:
  • Using the point \((4, 2)\):
  • Substitute into the equation: \( 2 = \frac{3}{4} \times 4 + b \)
  • Solve for \( b \): \( 2 = 3 + b \), thus \( b = -1 \)
This verifies that the y-intercept is \(-1\), confirming the line crosses the y-axis at that point.

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