Unit 13.2 Solving Systems of Equations Using Substitution
Substitution Method
The substitution method is one of the techniques used to solve a system of equations. This method involves solving one of the equations for one variable in terms of the other, and then substituting this expression into the other equation. This allows you to solve for one variable first, and then use this solution to find the other variable.
Steps to Solve a System of Equations Using the Substitution Method
- Solve One Equation for One Variable:
- Choose one of the equations and solve it for one of the variables in terms of the other variable. It’s often easiest to choose the equation and variable that will result in the simplest expression.
- Substitute the Expression into the Other Equation:
- Substitute the expression found in step 1 into the other equation. This will result in an equation with only one variable.
- Solve the Resulting Equation:
- Solve the equation obtained in step 2 to find the value of the first variable.
- Substitute Back to Find the Second Variable:
- Substitute the value of the first variable back into one of the original equations to find the value of the second variable.
- Check the Solution:
- Substitute the values of both variables into the original equations to ensure they satisfy both equations.
Solving systems of equations using substitution
Solve the systems of equations
2x + y = 7 (eq.1)
x – 2y = -3 (eq. 2)
Solve one of the equations for x or y. It does not matter which equation you choose or which variable you choose first
Equation 1:
y = 7 – 2x
Since we used equation 1 to solve for y, we need to plug that into equation 2. Plug in 7 – 2x anywhere there is a “y” in equation 2.
x – 2(7-2x) = -3
Now that we only have one variable, we can solve for x
x – 14 + 4x = -3
5x – 14 = -3
5x = 11
x = [latex]\frac{11}{5}[/latex]
We have a numerical value for x, now we plug that back into one of the original equations to solve for y
Equation 2:
[latex]\frac{11}{5}[/latex] – 2y = -3
– 2y = [latex]\frac{-26}{5}[/latex]
y = [latex]\frac{13}{5}[/latex]
Our solution set is ([latex]\frac{11}{5}[/latex] , [latex]\frac{13}{5}[/latex]). Plug both of these values into our original equations to check our work
2x + y = 7
2([latex]\frac{11}{5}[/latex]) + [latex]\frac{13}{5}[/latex] = 7
True
x – 2y = -3
[latex]\frac{11}{5}[/latex] – 2([latex]\frac{13}{5}[/latex]) = -3
True
This solution is valid