Question

Use linear combinations to solve the system of linear equations. $$\begin{aligned} &9 g-7 h=\frac{2}{3}\\\ &3 g+h=\frac{1}{3} \end{aligned}$$

Short answer

The solution to the system of equations is \(g = \frac{1}{10}, h = \frac{1}{15}\).

Step-by-step solution

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Rewrite the equations in their standard forms:\(9g - 7h = \frac{2}{3}\)\(3g + h = \frac{1}{3}\)

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Multiply the second equation by 7:\(9g - 7h = \frac{2}{3}\)\(21g + 7h = \frac{7}{3}\)

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Add the two equations together:\(30g = 3\)Then, divide the equation by 30 to find the value of \(g\):\(g = \frac{3}{30} = \frac{1}{10}\)

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Substitute the value of \(g\) into the second equation:\(3(\frac{1}{10}) + h = \frac{1}{3}\)Then, solve for \(h\):\(h = \frac{1}{3} - \frac{3}{10} = \frac{1}{15}\)

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Now that we found the values for \(g\) and \(h\), we can check them by substituting them back into the original equations to see if both sides of the equations balance