Question

Use linear combinations to solve the system of linear equations. $$\begin{aligned} &3 j+5 k=19\\\ &4 j-8 k=-4 \end{aligned}$$

Short answer

The solution to the system of equations is \(j = 3\) and \(k = 2\).

Step-by-step solution

Organize the system of equations

First, organize the system of equations in a more clearly readable form: \[ \begin{array}{ c l }3j + 5k & = 19 \4j - 8k & = -4\end{array} \]

Manipulate the equations using linear combination to eliminate one of the variables

Multiply the first equation by 4 and the second equation by 3 to make the \(j\) coefficients the same: \[ \begin{array}{ c l }12j + 20k & = 76 \12j - 24k & = -12\end{array} \] Then, subtract the second equation from the first one: \[ \begin{array}{ c l }(12j + 20k) - (12j - 24k) & = 76 - (-12)\end{array} \] This simplifies to: \[ \begin{array}{ c l }44k & = 88\end{array} \]

Solve for the remaining variable

Now that we have an equation with only one variable, we can easily solve for \(k\): \[ \begin{array}{ c l }k & = 88 / 44 \k & = 2\end{array} \]

Substitute the solved variable into one of the original equations

Now that we know \(k = 2\), we can substitute this value into one of the original equations to solve for \(j\). Substituting \(k = 2\) into the first equation, we get: \[ \begin{array}{ c l }3j + 5*(2) & = 19 \3j & = 19 - 10 \3j & = 9\end{array} \] Then, we can solve for \(j\): \[ \begin{array}{ c l }j & = 9 / 3 \j & = 3 \end{array} \]

Check the solutions

It's always a good idea to check the solutions by substitituted them back into the original equations to make sure the left and right sides match. Doing this we see: \[ \begin{array}{ c l }3*3 + 5*2 & = 19, \4*3 - 8*2 & = -4\end{array} \] Confirmation that both of these equations are true verifies that our solutions are correct.