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Chapter 7: Problem 65

Find the slope of each line, and sketch its graph. $$ x+2 y=4 $$

Short Answer

Expert verified
The slope is \[ -\frac{1}{2} \].

Step by step solution

01

- Rewrite the Equation in Slope-Intercept Form

The slope-intercept form of a line is given by the equation \[ y = mx + b \]where \( m \) is the slope and \( b \) is the y-intercept. Start with the given equation \[ x + 2y = 4 \]and solve for \( y \) to convert it into slope-intercept form.
02

- Isolate y

To isolate \( y \), first subtract \( x \) from both sides of the equation: \[ 2y = -x + 4 \]Next, divide each term by 2:\[ y = -\frac{1}{2}x + 2 \]Now, the equation is in the slope-intercept form \( y = mx + b \).
03

- Identify the Slope and Y-Intercept

From the equation \( y = -\frac{1}{2}x + 2 \), it is clear that the slope \( m \) is \[ -\frac{1}{2} \] and the y-intercept \( b \) is 2.
04

- Sketch the Graph

To sketch the graph, start by plotting the y-intercept (0, 2) on the coordinate plane. From this point, use the slope to find another point on the line. Since the slope is \[ -\frac{1}{2} \], it means you go down 1 unit (rise) and right 2 units (run) from the y-intercept. Plot this second point. Draw a straight line through the two points to complete the graph.

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Key Concepts

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

Slope
The slope of a line, often represented by the letter \( m \), measures its steepness and direction. This crucial concept tells us how much the \( y \)-coordinate of a point on the line changes per unit change in the \( x \)-coordinate. For example, a slope of \( -\frac{1}{2} \) means that for each step of 1 unit to the right (along the \( x \)-axis), the \( y \)-coordinate decreases by 0.5 units. Understanding slope helps in predicting the direction and inclination of the line on a graph. Without a proper grasp of slope, one cannot accurately sketch or understand linear equations.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), the \( b \) value represents the y-intercept. For the equation \( y = -\frac{1}{2}x + 2 \), the y-intercept is 2, meaning the line crosses the y-axis at the point (0, 2). This intercept is essential for graphing because it gives a fixed starting point to plot the line.
Graphing Linear Equations
Graphing linear equations involves plotting points and drawing a line that passes through them. To graph \( y = -\frac{1}{2}x + 2 \):
  • First, plot the y-intercept (0, 2) on the graph.
  • Next, use the slope to find another point. From (0, 2), move down 1 unit (since the slope is negative) and right 2 units.
  • Plot this second point at (2, 1).
  • Finally, draw a straight line through the two points to represent the equation.
By following these steps, you can accurately visualize the relationship between \( x \) and \( y \).
Isolating Variables
Isolating variables is a fundamental technique in algebra used to solve for one variable in terms of others. To rewrite \( x + 2y = 4 \) in slope-intercept form:
  • Start with the original equation: \( x + 2y = 4 \).
  • Subtract \( x \) from both sides to get \( 2y = -x + 4 \).
  • Divide every term by 2 to isolate \( y \): \( y = -\frac{1}{2}x + 2 \).
This isolates \( y \) on one side, converting the equation into the useful slope-intercept form. Mastering this skill is essential for solving and understanding various algebraic equations.

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Most popular questions from this chapter

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