Question

If \(6 k-5 l=27\) and \(3 l-2 k=-13\) and \(5 k-5 l=j,\) what is the value of \(j ?\)

Short answer

The value of \(j\) is 42.5.

Step-by-step solution

Rewrite the equations

First, rewrite the equations in a consistent order, either \(kl\) or \(lk\). In this case, let's choose \(kl\): \[2k + 3l = -13\] \[6k - 5l = 27 \]

Multiply to eliminate one variable

Multiply the first equation by 2 and the second equation by 3 so that the coefficient of \(k\) is the same in both equations. This will give: \[4k + 6l = -26\] \[18k - 15l = 81 \]

Sum both equations

Next, add both equations to eliminate one variable, l in this case. This gives \(22k = 55\).

Solve for k

Solve for \(k\) to find \(k = 55 / 22 = 2.5\). This is the value of \(k\).

Substitute value of k into one of the original equations

Substitute \(k = 2.5\) into the first original equation \(2k + 3l = -13\). This gives \(5 + 3l = -13\), hence \(3l = -18\) and \(l = -18 / 3 = -6\). This is the value of \(l\).

Substitute values of k and l into the equation for j

Substitute \(k = 2.5\) and \(l = -6\) into the equation \(5k - 5l = j\) to find \(j\). This gives \(j = 5(2.5) + 5(6) = 12.5 + 30 = 42.5\).