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Chapter 5: Problem 132

In the following exercises, solve the systems of equations by substitution. $$ \left\\{\begin{array}{l} 3 x-2 y=1 \\ -x+2 y=9 \end{array}\right. $$

Short Answer

Expert verified
(x, y) = (5, 7)

Step by step solution

01

Isolate One Variable

Choose one of the equations and solve for one of the variables. Here, solve for x in the second equation: -x + 2y = 9 Add x to both sides: 2y = x + 9 Subtract 9 from both sides: x = 2y - 9
02

Substitute the Expression

Substitute the expression for x from Step 1 into the first equation: 3(2y - 9) - 2y = 1 Distribute 3 in the equation: 6y - 27 - 2y = 1
03

Simplify and Solve for y

Combine like terms in the equation: (6y - 2y) - 27 = 1 4y - 27 = 1 Add 27 to both sides: 4y = 28 Divide both sides by 4: y = 7
04

Solve for x

Use the value of y in the expression for x found in Step 1: x = 2y - 9 Substitute y = 7: x = 2(7) - 9 x = 14 - 9 x = 5
05

Verify the Solution

Substitute x = 5 and y = 7 into the original equations to verify the solution: 3x - 2y = 1 3(5) - 2(7) = 1 15 - 14 = 1 1 = 1 (True) Second equation: -x + 2y = 9 -(5) + 2(7) = 9 -5 + 14 = 9 9 = 9 (True) Therefore, the solution (x, y) = (5, 7) satisfies both equations.

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Key Concepts

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

Substitution Method
The substitution method is a powerful technique for solving systems of equations. To begin, you need to isolate one variable in one of the equations. This means expressing one variable in terms of the other variable.
Once you have isolated the variable, you substitute this expression into the other equation. This substitution transforms the system into a single equation with one variable, making it easier to solve.
  • Choose one equation.
  • Rearrange it to isolate one variable.
  • Substitute this expression into the other equation.
This method can simplify complex problems, and practicing it often will improve your problem-solving skills.
Isolate Variable
Isolating a variable means rearranging the equation so that one variable stands alone on one side of the equation. In the example from the exercise, to isolate x in the equation -x + 2y = 9, you do the following:
  • First, add x to both sides: 2y = x + 9.
  • Then subtract 9 from both sides to solve for x: x = 2y - 9.

Isolating a variable is crucial because it creates an expression that can be substituted into the other equation. This step significantly simplifies solving systems of equations.
Verify Solution
After finding the values of the variables, it's essential to verify that these values satisfy both original equations. Verification ensures that our solution is correct and complete. To verify the solution:
  • Substitute the values of the variables back into the original equations.
  • Check if the left side equals the right side in both equations.

In our exercise, we found that x = 5 and y = 7. By substituting these values back into the equations: 3(5) - 2(7) = 1 and -(5) + 2(7) = 9.Both equations hold true, confirming that the solution is correct. Verifying solutions prevents errors and builds confidence in your mathematical skills.

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Most popular questions from this chapter

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