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Chapter 7: Problem 19

Solve each system by the substitution method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}2 x-y=-5 \\ x+5 y=14\end{array}\right.\)

Short Answer

Expert verified
The solution to the system of equations is \{ x = -1.455, y = 2.09 \}

Step by step solution

01

Rewrite the First Equation for Easier Substitution

Rearrange the first equation (2x - y = -5) to solve for one variable. It's easier to solve for x: \(x = (y + 5) / 2\)
02

Substitute the Expression from Step 1 into the Second Equation

Substitute \(x = (y + 5) / 2\) into the second equation \(x+5 y=14\). This gives \( ((y + 5) / 2) + 5y = 14\)
03

Solve for y

After clearing the division by multiplying through by 2, we find \( y + 5 + 10y = 28 \). This simplifies to \( 11y = 23 \), so \(y = 23 / 11 = 2.09\) (keeping the answer to two decimal places)
04

Substitute y back into one of the Equations to Solve for x

Now substitute \(y = 2.09\) back into the first original equation \(2x - y = -5\). This gives \(2x - 2.09 = -5\), or \(2x = -2.91\), so \(x = -2.91 / 2 = -1.455\) (keeping the answer to three decimal places)
05

Check the Solutions

Substituting \(x = -1.455\) and \(y = 2.09\) back into both original equations should make both equations true. For \(2x - y = -5\), we have \(-2.91 - 2.09 = -5\), which is a valid statement. Similarly, for \(x+5 y=14\), we have \(-1.455 + (5 * 2.09) = 14\), which is also a correct statement. Therefore, these solutions are valid.

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