Question

Solve each system of equations by using substitution

$\begin{array}{c}4c+2d=10\\ c+3d=10\end{array}$

Short answer

The solution of the system of equations is $\left(1 , 3\right)$.

Step-by-step solution

Step-1 – Apply the substitution method of solving equations

The algebraic method of substitution involves solving the one of the two equations for one variable in terms of other variable and then substituting the expression so formed for the variable in the second equation.

Step-2 – Solving one equation for c in terms of d

To solve the equation $c+3d=10$ for cin terms of d, subtract 3d from both sides as shown below.

$\begin{array}{c}c+3d=10\\ c+3d-3d=10-3d\\ c=10-3d\end{array}$

Step-3 – Substitute the expression

Now, substitute $c=10-3d$in the first equation $4c+2d=10$ and solve for d.

$\begin{array}{c}4c+2d=10\\ 4\left(10-3d\right)+2d=10\\ 40-12d+2d=10\\ -10d=-30\end{array}$

Simplify it further as

$\begin{array}{c}-10d=-30\\ d=3\end{array}$

Thus, the value of d is $3$.

Step-4 – Substitute the value of variable

To find the value of c, substitute $d=3$in the equation $c+3d=10$ and then solve for c as shown.

$\begin{array}{c}c+3d=10\\ c+3\left(3\right)=10\\ c+9=10\\ c=1\end{array}$

Thus, the value of c is $1$.

Hence, the solution of the provided system of equations is $\left(1 , 3\right)$.