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Magazine Chapter 2: Problem 53
Graph each equation \(5 x-4 y=13\)
Short Answer
Expert verified
Graph the line through the points \((0, -3.25)\) and \((4, 1.75)\).
Step by step solution
01
Rewrite the Equation in Slope-Intercept Form
Start by rewriting the given equation in the slope-intercept form, which is \( y = mx + b \). The given equation is \( 5x - 4y = 13 \). First, isolate \( y \) on one side:\[ 5x - 4y = 13 \]Subtract \( 5x \) from both sides:\[ -4y = -5x + 13 \]Divide each term by \(-4\) to solve for \( y \):\[ y = \frac{5}{4}x - \frac{13}{4} \]Now, the equation is in slope-intercept form: \( y = \frac{5}{4}x - \frac{13}{4} \).
02
Identify the Slope and Y-Intercept
From the slope-intercept form \( y = \frac{5}{4}x - \frac{13}{4} \), identify the slope and y-intercept:- The slope \( m \) is \( \frac{5}{4} \).- The y-intercept \( b \) is \( -\frac{13}{4} \) or \(-3.25\).
03
Plot the Y-Intercept on the Graph
Start by plotting the y-intercept on the graph. The y-intercept is the point where the graph intersects the y-axis; it is \((0, -3.25)\). Locate \(-3.25\) on the y-axis and place a point there.
04
Use the Slope to Find Another Point
Use the slope \( \frac{5}{4} \) to find another point on the line. The slope \( \frac{5}{4} \) means that for every 5 units you move up (rise), you move 4 units to the right (run). Starting from the y-intercept \((0, -3.25)\), move up 5 units and right 4 units to plot the next point at \((4, 1.75)\).
05
Draw the Line
With both points plotted, use a ruler to draw a straight line through the points \((0, -3.25)\) and \((4, 1.75)\). Extend the line in both directions, and add arrows to indicate it continues infinitely.
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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 is a standard way of writing equations to make them easier to graph. This format is expressed as \( y = mx + b \), where:
- \( m \) represents the slope of the line, indicating how steep the line is.
- \( b \) indicates the y-intercept, which is where the line crosses the y-axis.
Slope and Y-Intercept
Understanding the slope and y-intercept can greatly simplify the process of graphing a linear equation. The slope \( m \) and the y-intercept \( b \) in the equation \( y = mx + b \) play crucial roles:
- The slope \( \frac{5}{4} \) means that for each step you take horizontally (4 steps right), you rise by 5 units vertically. This gives the line its angle and direction.
- The y-intercept \( -\frac{13}{4} \) indicates where your line will intersect the y-axis. It's a starting point when you begin plotting.
Plotting Points on a Graph
To visualize the equation on a graph, start with the y-intercept \( (0, -3.25) \). This point is where the line crosses the y-axis. For this example, locate the point \( -3.25 \) along the vertical axis and mark it.Next, use the slope of \( \frac{5}{4} \) to determine another point. Begin at the y-intercept, then:
- Move up 5 units (this is the rise).
- Move right 4 units (this is the run).
Drawing a Straight Line
With two plotted points, drawing a straight line becomes straightforward. These points are like anchor points, and your task is to draw a single line through them:
- Use a ruler or any straight edge to ensure your line is precise and not wobbly. Align the ruler with both points \( (0, -3.25) \) and \( (4, 1.75) \).
- Draw a line through these points; this is your graph of the equation.
- Extend this line across your graph, adding arrows at both ends to show it continues indefinitely.
