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Chapter 7: Problem 6

Graph each equation using the slope and \(y\) -intercept. $$x-2 y=4$$

Short Answer

Expert verified
Graph the line through points (0, -2) and (2, -1) after rewriting as \(y = \frac{1}{2}x - 2\).

Step by step solution

01

Rewrite the Equation in Slope-Intercept Form

The slope-intercept form of an equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Start with the equation \(x - 2y = 4\). To rewrite it in slope-intercept form, solve for \(y\) by isolating it on one side of the equation. Subtract \(x\) from both sides:\[-2y = -x + 4\] Next, divide every term by -2:\[y = \frac{1}{2}x - 2\] Now, the equation is in slope-intercept form with \(m = \frac{1}{2}\) and \(b = -2\).
02

Identify the Slope and Y-Intercept

From the equation \(y = \frac{1}{2}x - 2\), we can see the slope \(m\) is \(\frac{1}{2}\) and the y-intercept \(b\) is \(-2\).This means the graph of the line will cross the y-axis at \(-2\) and will rise 1 unit for every 2 units it moves to the right.
03

Plot the Y-Intercept

Start by plotting the y-intercept on the graph. Since \(b = -2\), place a point on the y-axis at \(y = -2\). This is the point \( (0, -2) \).
04

Use the Slope to Find Another Point

Use the slope \(m = \frac{1}{2}\) to find another point on the line. The slope \(\frac{1}{2}\) indicates you rise 1 unit up for every 2 units you move right.Starting at the y-intercept (0, -2), move up 1 unit to \(y = -1\), and 2 units to the right to \(x = 2\), placing a point at (2, -1).
05

Draw the Line

Draw a straight line through the points (0, -2) and (2, -1). Extend the line in both directions and add arrows to indicate it extends infinitely.

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Key Concepts

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

Graphing Linear Equations
Graphing linear equations involves visualizing algebraic expressions on a coordinate plane. This is crucial in understanding how mathematical relationships look and function in a 2D space. For linear equations such as \(y = \frac{1}{2}x - 2\), we first need to rewrite them into the slope-intercept form. This form, \(y = mx + b\), allows easy identification of fundamental properties like slope and y-intercept.
  • Begin with writing the equation in slope-intercept form by solving for \(y\).
  • Identify the slope \(m\) and y-intercept \(b\).
  • Plot these points and use the slope to find additional points.
This process helps in plotting the equation accurately on the graph, making it clear how changes in the equation reflect geometrically.
Y-Intercept
The y-intercept is a key element in graphing linear equations; it marks where the line crosses the y-axis. In the equation \(y = mx + b\), \(b\) represents the y-intercept.
For the line described by \(y = \frac{1}{2}x - 2\), the y-intercept is at \(b = -2\). This means the line crosses the y-axis at the point \((0, -2)\).
  • Simply locate \(y = -2\) on the y-axis.
  • Place a point at \((0, -2)\) to set the starting point of your line.
The y-intercept provides an anchor point from which the slope can help find additional points on the line. Without plotting this crucial intercept, depicting the entire line accurately becomes challenging.
Slope Calculation
Slope calculation is essential for determining a line's steepness and direction. The slope indicates how quickly or slowly a line ascends or descends when moving from left to right.
For the equation \(y = \frac{1}{2}x - 2\), the slope \(m\) is \(\frac{1}{2}\). This tells us:
  • The line rises 1 unit for every 2 units it moves to the right.
  • A positive slope means the line ascends as it moves from left to right.
To practically apply that slope, start at the y-intercept. Move up by the numerator (rise), and right by the denominator (run). The precision in calculating slope ensures accurate plotting and representation of a line graph. Understanding slope helps in predicting and extrapolating the line鈥檚 direction and pace across the graph.

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