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Chapter 7: Problem 36

Solve each system by the addition method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}5 x=4 y-8 \\ 3 x+7 y=14\end{array}\right.\)

Short Answer

Expert verified
The solution to the system of equations is \( x = 4 \) and \( y = 46/47 \).

Step by step solution

01

Arrange the equations

Rearrange the equations in the same order of their variables, making sure that the coefficients of x in the first equation and y in the second equation are equal. So, we get the system in the form: \(\left\{\begin{array}{l}5 x-4 y=-8 \ 3 x-7 y=14\end{array}\right.\).
02

Multiply and add

Multiply first equation by 3 and second equation by 5 to make the coefficients of x the same. Then add the equations together to eliminate x. This gives: \(15x - 12y = -24\) and \(15x - 35y = 70\). Adding these two expressions gives \( -47y = 46 \).
03

Solve for y

Divide the equation obtained in step 2 by the coefficient of y to obtain the value of y: \( y = -46/-47 = 46/47 \).
04

Substitute y into the original equation

Now, substitute y = 46/47 into the first original equation \(5x = 4y - 8\), which becomes \(5x = 4*(46/47) - 8\).
05

Solve for x

To get x alone, divide both sides of the simplified equation from step 4 by 5: \( x = [4*(46/47) - 8] / 5 \). After simplifying the equation, \( x = 4 \).
06

Check the solution

Substitute x = 4 and y = 46/47 into the original equations to verify that they are the correct solutions. After substitution, the left and right sides of the equations should be equal.

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