Q1. - Algebra 2

Question

Give an example of a system of equations that is more easily solved by substitution and a system that is more easily solved by elimination.

Step-by-Step Solution

Answer

The system of equations that can be solved by substitution is provided below.

$\begin{array}{c}y=3+x\\ x+5y=1\end{array}$

The system of equations that can be solved by substitution is provided below.

$\begin{array}{c}5x+7y=10\\ x-y=2\end{array}$

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 y in terms of x

Consider the system of equations

$\begin{array}{c}y=3+x\\ x+5y=1\end{array}$

To solve the equation $y=3+x$ for y in terms of x, observe that the equation is already in the form.

Step-3 – Substitute the expression

Now, substitute $y=3+x$ in the equation $x+5y=1$ and solve for x.

$\begin{array}{c}x+5\left(x+3\right)=1\\ x+5x+15=1\\ 6x+15=1\\ 6x=-14\end{array}$

Simplify it further as.

$\begin{array}{c}6x=-14\\ x=\frac{-14}{6}\\ x=\frac{-7}{3}\end{array}$

Step-4 – Substitute the value of variable

To find the value of y, substitute $x=\frac{-7}{3}$ in the equation $y=3+x$ and then solve for y as shown.

$\begin{array}{c}y=3-\frac{7}{3}\\ y=\frac{9-7}{3}\\ y=\frac{2}{3}\end{array}$

Thus, the value of y is $\frac{2}{3}$.

Hence, the solution of the provided system of equations is $\left(\frac{-7}{3},\frac{2}{3}\right)$.

Step-5 – 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-6 – Multiplying the equation by a nonzero number

Consider the system of equations

$\begin{array}{c}5x+7y=10\\ x-y=2\end{array}$

To solve the equations, multiply $x-y=2$ by 7 and add the resulting equation and the other equation as shown below.

$7x-7y=14$

Step-7 – Adding/Subtracting the equations

Now, add $7x-7y=14$ and the equation $5x+7y=10$ and solve.

$\begin{array}{ccccccc}& 5x& +& 7y& =& & 10\\ & 7x& -& 7y& =& & 14\\ +& & +& & & +& \\ & 12x& +& 0& =& & 24\end{array}$

Simplify it further as

$\begin{array}{c}12x=24\\ x=2\end{array}$

Thus, the value of x is $2$.

Step-8 – Substitute the value of variable

To find the value of y, substitute $x=2$ in the equation $x-y=2$ and then solve as shown.

$\begin{array}{c}x-y=2\\ 2-y=2\\ y=0\end{array}$

Thus, the value of y is 0.

Hence, the solution of the provided system of equations is $\left(2,0\right)$.