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Chapter 2: Problem 38

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ x-2 y=-4 $$

Short Answer

Expert verified
x-intercept: (-4, 0); y-intercept: (0, 2)

Step by step solution

01

- Find the x-intercept

To find the x-intercept, set y to 0 in the equation and solve for x. Starting with the equation: \[ x - 2y = -4 \] Set y = 0: \[ x - 2(0) = -4 \] Simplify to get: \[ x = -4 \] Therefore, the x-intercept is at the point (-4, 0).
02

- Find the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y. Starting with the equation: \[ x - 2y = -4 \] Set x = 0: \[ 0 - 2y = -4 \] Simplify to get: \[ -2y = -4 \] Divide both sides by -2: \[ y = 2 \] Therefore, the y-intercept is at the point (0, 2).
03

- Plot the intercepts on a graph

Plot the points (-4, 0) and (0, 2) on a coordinate plane. These points represent the x-intercept and y-intercept, respectively.
04

- Draw the line

Draw a straight line through the points (-4, 0) and (0, 2). This line represents the equation \( x - 2y = -4 \).

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Key Concepts

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

Linear Equations
A linear equation is an algebraic expression that forms a straight line when graphed on a coordinate plane. Typically, linear equations are written in the form of:
\[ax + by = c\]
where \(a\), \(b\), and \(c\) are constants. In our specific example, the equation is \(x - 2y = -4\).
Linear equations have important properties:
  • They represent straight lines.
  • The graph has a constant slope.
  • There are no exponents or powers higher than 1 on any variable.
Understanding linear equations is crucial when dealing with intercepts and graphing.
Graphing
Graphing is the method of plotting points on a coordinate plane to represent an equation visually. For linear equations, the process involves a few steps:
  • First, find the x-intercept and y-intercept. These are points where the line crosses the x-axis and y-axis, respectively.
  • Plot these intercepts on the coordinate plane.
  • Draw a straight line through the plotted points.
In the given exercise, the x-intercept is found by setting \(y\) to 0, and the y-intercept by setting \(x\) to 0. By plotting the points (-4, 0) and (0, 2), and drawing a line through them, we graph the equation \(x - 2y = -4\).
Coordinate Plane
A coordinate plane is a two-dimensional plane formed by the intersection of a vertical line (y-axis) and a horizontal line (x-axis). These axes divide the plane into four quadrants:
  • Quadrant I - Positive x and y values
  • Quadrant II - Negative x and positive y values
  • Quadrant III - Negative x and y values
  • Quadrant IV - Positive x and negative y values
Points on the plane are identified by ordered pairs \((x, y)\). For example, the point (-4, 0) lies on the x-axis while (0, 2) lies on the y-axis. By understanding the coordinate plane, students can correctly place points and graph linear equations.

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Most popular questions from this chapter

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