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Chapter 7: Problem 28

Solve each system by the addition method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}3 x+2 y=14 \\ 3 x-2 y=10\end{array}\right.\)

Short Answer

Expert verified
The solution to the system of equations is \( x = 4, y = 1 \)

Step by step solution

01

Add the Equations

Add the two equations to eliminate the \( y \) term: \[ (3x + 2y) + (3x - 2y) = 14 + 10 \]. This simplifies to: \[ 6x = 24 \]
02

Solve for x

Divide both sides by 6 to solve for \( x \): \[ x = 24/6 \]. This equation simplifies to: \[ x = 4 \]
03

Substitute x into the First Equation

Substitute \( x = 4 \) into the first equation: \[ 3(4) + 2y = 14 \]. This simplifies to: \[ 12 + 2y = 14 \]
04

Solve for y

Subtract 12 from both sides: \[ 2y = 14 - 12 \]. This simplifies to: \[ 2y = 2 \]. Divide both sides by 2 to solve for \( y \): \[ y = 2/2 \]. This equation simplifies to: \[ y = 1 \]
05

Check the Solution

This step involves plugging the solution set \( x = 4, y = 1 \) into the original set of equations to ensure that they are the correct solutions. \[ 3x + 2y = 14 \] becomes \[ 3(4) + 2(1) = 14 \] which simplifies to \[ 14 = 14 \], and \[ 3x - 2y = 10 \] becomes \[ 3(4) - 2(1) = 10 \] which simplifies to \[ 10 = 10 \]. In both cases, the equations hold true, indicating that the solution set is correct.

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