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Chapter 5: Problem 57

Review graphing linear equations (Chapter 3). Graph. $$3 x-4 y=12[3.3]$$

Short Answer

Expert verified
The slope is \(\frac{3}{4}\) and the y-intercept is -3. Graph the line passing through (0, -3) and (4, 0).

Step by step solution

01

Rewrite the equation in slope-intercept form

Start with the given equation: \(3x - 4y = 12\). To rewrite it in the form \(y = mx + b\), solve for \(y\).Subtract \(3x\) from both sides:\[-4y = -3x + 12\]Next, divide both sides by -4:\[y = \frac{3}{4}x - 3\]
02

Identify the slope and y-intercept

From the equation \(y = \frac{3}{4}x - 3\), identify the slope \(m\) and y-intercept \(b\).Slope (\(m\)): \(\frac{3}{4}\)Y-intercept (\(b\)): -3
03

Plot the y-intercept

Locate the y-intercept on the graph. For \(b = -3\), place a point at (0, -3) on the y-axis.
04

Use the slope to find another point

The slope \(\frac{3}{4}\) means you rise 3 units up and run 4 units to the right from the y-intercept. Starting at (0, -3), move up 3 units to (0, 0), then 4 units to the right to (4, 0). Mark the point (4, 0).
05

Draw the line

Plot both points (0, -3) and (4, 0) on the graph. Draw a straight line through these points to represent the equation.

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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 of a linear equation makes it easy to graph a line. The general format is: \( y = mx + b \). In this equation:
  • \( y \): Represents the dependent variable.
  • \( x \): Represents the independent variable.
  • \( m \): Represents the slope of the line.
  • \( b \): Represents the y-intercept of the line.
Using slope-intercept form simplifies plotting because it directly shows the slope and the y-intercept.
y-intercept
The y-intercept is where the line crosses the y-axis. In the equation \( y = \frac{3}{4}x - 3 \), the y-intercept is -3. This means:
  • The line touches the y-axis at the point (0, -3).
The y-intercept is crucial for graphing because it gives you a starting point for the line. From here, you can use the slope to find other points.
slope
The slope of a line shows how steep it is. Slope is defined as the 'rise over run' between two points. In our equation \( y = \frac{3}{4}x - 3 \), the slope \( m \) is \( \frac{3}{4} \). This means:
  • For every 4 units you move to the right (run), you move 3 units up (rise).
To use the slope:
1. Start at the y-intercept (0, -3).
2. Move 3 units up to (0, 0).
3. Move 4 units right to (4, 0).
Mark the point (4, 0) and draw a line through both points to graph the equation.

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