Problem 17 Solve each system by the substit... [FREE SOLUTION]

Chapter 7: Problem 17

Solve each system by the substitution method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}x+3 y=5 \\ 4 x+5 y=13\end{array}\right.\)

Short Answer

Expert verified

The solutions for the system of equations are \(x = 2\) and \(y = 1\), which have been validated by substitution into both original equations.

Step by step solution

Isolate one variable

Select one equation and solve for one variable. Let's choose the first equation and solve for \(x\). That gives us: \(x = 5 - 3y\).

Substitute the expression

Substitute the expression for \(x\) found in step 1 into the second equation to solve for \(y\). That results in: \(4(5 - 3y) + 5y = 13\). Simplification gives: \(20 - 12y + 5y = 13\). Solving for \(y\) gives \(y = 1\).

Solve for the other variable

Now, substitute \(y = 1\) into the equation \(x = 5 - 3y\), which was given in the first step for \(x\). That results in: \(x = 5 - 3*1\), hence \(x = 2\).

Check the solution

Finally, substitute \(x = 2\) and \(y = 1\) into both original equations to ensure they satisfy both equations. \[\left\{\begin{array}{l}2+3*1=5 \ 4*2+5*1=13\end{array}\right.\] Both equations are satisfied by \(x = 2\) and \(y = 1\), so these are the solutions.

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Solve each system by the substitution method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}x+3 y=5 \\ 4 x+5 y=13\end{array}\right.\)

Short Answer

Step by step solution

Isolate one variable

Substitute the expression

Solve for the other variable

Check the solution

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