Answer
The solution to this system of equations is $k = 1$ and $t = 4$.
Work Step by Step
When we use elimination to solve a system of equations, we need to make sure that one of the variables in both equations is the same but differing only in sign.
Let's take a look at the system of equations in this exercise. In this case, we want to convert the equations such that one variable term in both equations is the same value but with opposite signs. We can do this by multiplying the first equation by $5$ and the second equation by $3$:
$15t + 40k = 100$
$-15t - 18k = -78$
Let's add the two equations:
$22k = 22$
Divide each side by $22$ to solve for $k$:
$k = 1$
Substitute this value for $k$ into one of the equations to solve for $t$:
$3t + 8(1) = 20$
Multiply first:
$3t + 8 = 20$
Collect constant terms on the right side of the equation:
$3t = 12$
Divide each side by $3$ to solve for $t$:
$t = 4$
The solution to this system of equations is $k = 1$ and $t = 4$.
To see if this solution is correct, substitute in the values we just found for $k$ and $t$ into one of the original equations:
$5(4) + 6(1) = 26$
Multiply first:
$20 + 6 = 26$
Subtract:
$26 = 26$
The two sides are equal; therefore, this solution is correct.
This system of equations is consistent because it has at least one solution. It is also independent because there is only one solution.
