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Chapter 4: Problem 71

For exercises 71-74, (a) solve by elimination. (b) check by graphing. $$ \begin{aligned} x+y &=12 \\ 2 x-y &=9 \end{aligned} $$

Short Answer

Expert verified
The solution is \( x = 7 \) and \( y = 5 \).

Step by step solution

01

- Align the Equations

First, write the given system of equations one under the other: \[ \begin{aligned} x + y &= 12 \ 2x - y &= 9 \ \text{(1)} & \text{(2)} \ \ \text{We'll use the elimination method to solve.} \]
02

- Adjust the Equations for Elimination

To eliminate one variable, add equation (1) to equation (2) to cancel out \( y \). \[ \begin{aligned} x + y & = 12 \ + \, 2x - y & = 9 \ \text{----------------} \ 3x & = 21 \ \ \text{Now, solve for } x. \]
03

- Solve for x

Now, solving the equation \( 3x = 21 \): \[ x = \frac{21}{3} \ x = 7 \ \ \text{Now that we have } x, \text{ we substitute it back into one of the original equations to find } y.\]
04

- Solve for y

Using the value of \( x \text{ from the equation } x + y = 12 \): \[ 7 + y = 12 \ y = 12 - 7 \ y = 5 \]
05

- Verify the Solution

Substitute \( x = 7 \ \text{ and } y = 5 \ \text{ back into the equations to verify.} \)Verify equation (1): \( 7 + 5 = 12 \ 12 = 12 \) Verify equation (2): \( 2(7) - 5 = 9 \ 14 - 5 = 9 \ \ \text{Since both sides of each equation are valid, the solution is verified.} \)
06

- Check by Graphing

Graph the lines of the equations to find their intersection.Graph equation (1): \( x + y = 12 \) Graph equation (2): \( 2x - y = 9 \)The intersection of the lines (7, 5) should confirm the solution.

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Key Concepts

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

Elimination Method
The elimination method is a powerful technique used to solve systems of linear equations. It works by canceling out one variable, making it easier to solve for the other. To start, align the equations one under the other. In our example:
  • \[ x + y = 12 \]
  • \[ 2x - y = 9 \]
Add these equations to eliminate \( y \):
  • \[ (x + y) + (2x - y) = 12 + 9 \]
This simplifies to:
  • \[ 3x = 21 \]
Solving for \( x \):
  • \[ x = 7 \]
Now use this \( x \) value to find \( y \). Substitute \( x = 7 \) into the original equation \( x + y = 12 \), getting:
  • \[ 7 + y = 12 \]
  • \[ y = 5 \]
So, the solution is \( x = 7 \) and \( y = 5 \).
Graphing Method
The graphing method provides a visual means to solve systems of equations. By graphing the two lines, their intersection point represents the solution. For our equations, graphing yields:
  • Equation (1): \( x + y = 12 \)
  • Equation (2): \( 2x - y = 9 \)
Graph these by first solving for \( y \):
  • Convert Equation (1): \( y = 12 - x \)
  • Convert Equation (2): \( y = 2x - 9 \)
Plot these two lines on a coordinate plane. The lines intersect at the point \( (7, 5) \), confirming that \( x = 7 \) and \( y = 5 \).This point is our solution, visually verifying the results obtained analytically.
Verification of Solutions
To ensure our solutions are correct, we verify them by substituting back into the original equations. For the solved values \( x = 7 \) and \( y = 5 \):
  • Substitute into Equation (1):\[ 7 + 5 = 12 \]
  • Equation (1): Check is correct as \( 12 = 12 \)
  • Substitute into Equation (2):\[ 2(7) - 5 = 9 \]
  • Equation (2): Check is correct as \( 14 - 5 = 9 \)
Both sides of the original equations balance out with our found values, confirming that our solution is indeed valid and accurate.
Intersection of Lines
The intersection of lines in a system of linear equations reveals the solution to the system. Visually, the point where the two lines meet is the solution. For our example:
  • Graph the line \( x + y = 12 \), which intersects the \( y \)-axis at \( y = 12 \) and has a slope of \(-1\).
  • Graph the line \( 2x - y = 9 \), which intersects the \( y \)-axis at \( y = -9 \) and has a slope of \( 2 \).
Where these two lines intersect (\( 7, 5 \)) is the solution, showing \( x = 7 \) and \( y = 5 \). This shows both the analytic and visual methods affirming each other.

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Most popular questions from this chapter

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