Unit 4, Lesson 13
Solving Systems by Substitution
Integrated Math 1
13.1
Warm-up
Find the value of the last line. Be prepared to explain your reasoning.
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Find the value of the last line. Be prepared to explain your reasoning.
Here are four systems of equations. Solve each system by first finding the value of one variable and then using it to find the value of any other variables. Then, check your solutions by substituting them into the original equations to see if the equations are true.
The solution to a system can usually be found by graphing, but graphing may not always be the most precise or the most efficient way to solve a system.
Here is a system of equations:
The graphs of the equations show an intersection at approximately 20 for and approximately 10 for .
Without technology, however, it is not easy to tell what the exact values are.
Instead of solving by graphing, we can solve the system algebraically. Here is one way.
If we subtract from each side of the first equation, , we get an equivalent equation: . Rewriting the original equation this way allows us to isolate the variable .
Because is equal to , we can substitute the expression in the place of in the second equation. Doing this gives us an equation with only one variable, , and makes it possible to find .
Now that we know the value of , we can find the value of by substituting 20.2 for in either of the original equations and solving the equation.
The solution to the system is the pair and , or the point on the graph.
This method of solving a system of equations is called solving by substitution, because we substituted an expression for into the second equation.
Substitution is the action of replacing a variable with a number or expression it is equal to.