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Chapter 7: Problem 35

Solve each system by the addition method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}2 x=3 y-4 \\ -6 x+12 y=6\end{array}\right.\)

Short Answer

Expert verified
The solution to the system is \(x = -5\) and \(y = 4\).

Step by step solution

01

Align the Variables

Firstly, arrange the given equations in the standard form where x and y terms are on the left-hand side and constants on the right-hand side. The rearranged form is: \(2x - 3y = -4\) and \(-6x + 12y = 6\).
02

Make Coefficients of y's the Same in Both Equations

To eliminate the variable y, make the coefficients of y's the same in both equations. Multiply the first equation by 4 to achieve this alignment. The transformed equations are: \(8x - 12y = -16\) and \(-6x + 12y = 6\).
03

Add the Two Equations to Eliminate y

Now, add the two transformed equations. This will give: \(2x = -10\).
04

Solve for x

Divide both sides of the equation \(2x = -10\) by 2 to solve for x. This yields \(x = -5\).
05

Substitute x into the Original Equation to Solve for y

Substitute \(x = -5\) into the original equation \(2x - 3y = -4\). Solving for y gives \(y = 4\).
06

Check the Solutions

Substitute the obtained solutions \(x = -5\) and \(y = 4\) into the original equations to verify them. They hold true, so the solutions are valid.

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