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Chapter 7: Problem 15

Solve each system by the substitution method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}x+3 y=8 \\ y=2 x-9\end{array}\right.\)

Short Answer

Expert verified
The solution to the system of equations is \(x = 5\) and \(y = 1\).

Step by step solution

01

Identify the Equation for Substitution

From the given system of equations, you can see that the second equation \(y = 2x - 9 \) is already solved for \(y\). So you should use this equation for substitution.
02

Perform the Substitution and Simplification

Substitute \( y = 2x - 9 \) into the first equation in place of \(y\). This gives: \(x + 3(2x - 9) = 8\). Simplify this equation: collect like terms to get \(7x - 27 = 8\).
03

Solve the Resulting Equation for \(x\)

Solving the equation \(7x - 27 = 8\) for \(x\) involves two steps: first add 27 to both sides of the equation to obtain \(7x = 35\), then divide both sides by 7 to solve for \(x\). This gives \(x = 5\).
04

Find the Value of \(y\)

Now that you have \(x = 5\), substitute this into \(y = 2x - 9\) to find the value of \(y\). This gives \(y = 2*5 - 9\), which simplifies to \(y = 1\).
05

Check the Solution

Check the solution by substituting \(x = 5\) and \(y = 1\) into the original equations to verify that both equations are satisfied. First equation: \(5 + 3*1 = 8\), which is satisfied. Second equation: \(1 = 2*5 - 9 = 1\), which is also satisfied. Therefore, the solution \(x = 5\) & \(y = 1\) is correct.

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