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Chapter 4: Problem 47

Solve by graphing. $$ \begin{aligned} &x+2 y=6 \\ &y=4 x-6 \end{aligned} $$

Short Answer

Expert verified
The intersection point is the solution. Plotting both equations shows they intersect at (2,1), so the solution is \(x=2\) and \(y=1\).

Step by step solution

01

- Write the equations in slope-intercept form

The given equations are: 1. \(x+2y=6\) 2. \(y=4x-6\). The second equation is already in slope-intercept form \(y=mx+b\). Rewrite the first equation: \(x+2y=6\) to \(2y=-x+6\) and then \(y=-\frac{1}{2}x+3\).
02

- Identify slopes and y-intercepts

For the first equation \(y=-\frac{1}{2}x+3\), the slope (m) is \(-\frac{1}{2}\) and the y-intercept (b) is 3. For the second equation \(y=4x-6\), the slope (m) is 4, and the y-intercept (b) is -6.
03

- Graph the first equation y=-\frac{1}{2}x+3

To graph \(y=-\frac{1}{2}x+3\), start at the y-intercept (0,3). From there, use the slope \(-\frac{1}{2}\) (down 1 unit and right 2 units) to plot additional points.
04

- Graph the second equation y=4x-6

To graph \(y=4x-6\), start at the y-intercept (0,-6). From there, use the slope 4 (up 4 units and right 1 unit) to plot additional points.
05

- Find the intersection point of the two lines

The intersection point of the two lines on the graph represents the solution to the system. After plotting both lines, find the point where they cross.
06

- Verify the solution

Ensure that the intersection point satisfies both original equations by substituting the coordinates of the intersection point back into the equations.

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Key Concepts

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

Slope-Intercept Form
Understanding the slope-intercept form is crucial for graphing linear equations. This form of a linear equation is \(y = mx + b\) where:
  • m is the slope of the line, showing how steep it is.
  • b is the y-intercept, the point where the line crosses the y-axis.

In the given exercise, the second equation \(y = 4x - 6\) is already in slope-intercept form. We rewrite the first equation \(x + 2y = 6\) as \(y = -\frac{1}{2}x + 3\) to match this format.
Graphing Linear Equations
Once we have the equations in slope-intercept form, we can graph them.
  • Start with the y-intercept (\(b\)). For \(y = -\frac{1}{2}x + 3\), begin at (0, 3). For \(y = 4x - 6\), start at (0, -6).
  • Use the slope (\(m\)). For \(y = -\frac{1}{2}x + 3\), the slope is -1/2. This means go down 1 unit and right 2 units. For \(y = 4x - 6\), slope is 4. Move up 4 units and right 1 unit.

Continue plotting points until the lines are clearly drawn on the graph.
Intersection Point
The intersection point of the two lines is where the lines meet on the graph. This point corresponds to the solution of the system of linear equations.
By plotting both lines accurately, the intersection point can be found visually. This point will have coordinates (x, y) that satisfy both equations.
Linear Systems
A system of linear equations consists of two or more linear equations.
  • The solution to a system is the set of values (x, y) that satisfy all equations in the system simultaneously.
  • Graphing is a useful method to visualize where the equations intersect.
    In this case, solving by graphing helps to find where \(y = -\frac{1}{2}x + 3\) and \(y = 4x - 6\) intersect.
Verification of Solutions
Once the intersection point is found on the graph, it is important to verify that it satisfies both original equations.
  • Substitute x and y values of the intersection back into both equations.
  • For example, if the intersection point is (2, 1), substitute x = 2 and y = 1 into \(x + 2y = 6\) and \(y = 4x - 6\).

If the point holds true for both, it confirms the solution is correct.

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Most popular questions from this chapter

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