Question

Solve each system of equations by using either substitution or elimination.

$\begin{array}{c}\frac{1}{4}x+y=\frac{7}{2}\\ x-\frac{1}{2}y=2\end{array}$

Short answer

The solution of the system of equations is $\left(\frac{30}{9} , \frac{24}{9}\right)$.

Step-by-step solution

Step 1. Apply the elimination method of solving equations

The algebraic method of elimination involves adding or subtracting the equations to eliminate one of the variables and forming new equation that is true. Sometimes, direct addition or subtraction of equations does not eliminate the variable then one equation requires formation of equivalent equation through multiplication so that one of the two variables has the same or opposite coefficient in both the equations. Multiplying the equation by a nonzero number, resulting new equation has same set of solutions.

Step 2. Multiplying the equation by a nonzero number

To solve the equations, multiply $\frac{1}{4}x+y=\frac{7}{2}$by 4 and $x-\frac{1}{2}y=2$ by 8 then add the resulting equations as shown below.

$\begin{array}{c}4\left(\frac{1}{4}x+y\right)=\left(4\right)\left(\frac{7}{2}\right)\\ x+4y=14\end{array}$

$\begin{array}{c}8\left(x-\frac{1}{2}y\right)=8\left(2\right)\\ 8x-4y=16\end{array}$

Step 3. Adding/Subtracting the equations

Now, add $x+4y=14$and $8x-4y=16$.

$\begin{array}{ccccccc}& x& +& 4y& =& & 14\\ & 8x& -& 4y& =& & 16\\ & & & & & & \\ & 9x& +& 0& =& & 30\end{array}$

Simplify it further as

$\begin{array}{c}9x=30\\ x=\frac{30}{9}\end{array}$

Thus, the value of x is $\frac{30}{9}$.

Step 4. Substitute the value of variable

To find the value of y, substitute $x=\frac{30}{9}$in the equation $x-\frac{1}{2}y=2$ and then solve as shown.

$\begin{array}{c}x-\frac{1}{2}y=2\\ \frac{30}{9}-\frac{1}{2}y=2\\ \frac{60-9y}{18}=2\\ 60-9y=36\end{array}$

Simplify it further as

$\begin{array}{c}60-9y=36\\ -9y=-24\\ y=\frac{24}{9}\end{array}$

Thus, the value of y is $\frac{24}{9}$.

Hence, the solution of the provided system of equations is $\left(\frac{30}{9} , \frac{24}{9}\right)$.