How can we use the concept:
To solve a system of equations by substitution you can use the following steps:
- Solve one of the equations for a variable
- This can be chosen by you, usually we try to solve for a variable that has a coefficient of one.
- Substitute the expression from step 1 into the second equation
- Solve the resulting equation
- Note that at this point you should only have one variable!
- Substitute the solution from the previous step back into either of the original equations to solve for the other variable.
- Write the solution as an ordered pair.
Special Cases:
- Infinite Number of solutions: Two lines that are coinciding have infinitely many solutions In the process of substitution, this occurs when in step 3, we begin to solve the equation and the variables cancel out. The remaining equality is a true statement.
- No solution: two lines that are parallel will never touch resulting in no solutions. In the process of substitution, this occurs when in step 3, we begin to solve the equation and the variables cancel out. The remaining equality is not a true statement
Sample Math Problems
Question 1:
Solve the system.
5x - 8y = 1
x + 2y = 11
Answer:
Step 1: Solve one of the equations for a variable.
Using the second equation, we can solve for x since it is the only variable with a coefficient of 1.
x + 2y = 11
-2y -2y
_____________
x = 11 - 2y
Step 2: Substitute the expression from step 1 into the second equation.
5(11 - 2y) - 8y = 1
Step 3: Solve the resulting equation.
5(11 - 2y) - 8y = 1
55 - 10y - 8y = 1
55 - 18y = 1
-18y = -54
y = 3
Step 4: Substitute the solution from the previous step back into either of the original equations to solve for the other variable.
5x - 8(3) = 1 OR x + 2(3) = 11
5x - 24 = 1 x + 6 = 11
5x = 25 x = 5
x = 5
Step 5: Write the solution as an ordered pair.
Since x = 5 and y = 3, the solution is (5, 3)
Question 2:
-2x + 8y = -12
4x + 2y = -12
Answer:
Step 1: Solve one of the equations for a variable. Try to choose one that will not create fractions!
4x + 2y = -12
-4x -4x
_______________
2y = -12 - 4x
y = -6 - 2x
Step 2: Substitute the expression from step 1 into the second equation.
Step 3: Solve the resulting equation.
-2x + 8(-6 - 2x) = -12
-2x - 48 - 16x = -12
-18x - 48 = -12
-18x = 36
x = -2
Step 4: Substitute the solution from the previous step back into either of the original equations to solve for the other variable.
-2(-2) + 8y = -12 OR 4(-2) + 2y = -12
4 + 8y = -12 -8 + 2y = -12
8y = -16 2y = -4
y = -2 y = -2
Step 5: Write the solution as an ordered pair.
Since x = -2 and y = -2, the solution is (-2, -2)
Question 3:
6x + 2y = -32
3x + y = -16
Answer:
Step 1: Solve one of the equations for a variable.
3x + y = -16
-3x -3x
_____________
y = -16 - 3x
Step 2: Substitute the expression from step 1 into the second equation.
Step 3: Solve the resulting equation.
6x + 2(-16 - 3x) = -32
6x - 32 - 6x = -32
-32 = -32
Since the variables cancel out and this is a true statement, that means the equations represent the same lines. There are infinitely many solutions.
Question 4: Nicole is trying to determine which cell phone plan for her to purchase. Company A charges a flat fee of $35 plus $0.22 per megabyte of data used. Company B charges a $15 flat fee plus $0.27 per megabyte of data used. How many megabytes (m) of data should Nicole use so that the cost of her bill (b) would be equal? How much would that bill cost?
Answer:
We need to create an equation for both companies. The cost per megabyte would be the rates of change, or the slope, and the flat fee would be the initial amount or y-intercept.
We need to create an equation for both companies. The cost per megabyte would be the rates of change, or the slope, and the flat fee would be the initial amount or y-intercept.
Company A: b = 0.22m + 35
Company B: b = 0.27m + 15
We can now solve the system. Since both equations are set equal to “b” we can substitute one equation into the other, resulting in this:
0.22m + 35 = 0.27m + 15
We can now move to step 3 of solving the equation.
0.22m + 35 = 0.27m + 15
-0.22m -0.22m
________________________
35 = 0.05m + 15
-15 -15
_________________
20 = 0.05m
400 = m
Step 4: substitute back into either original equation
b = 0.22(400) + 35
b = 123
Step 5: Since this is a word problem, we answer the question in a sentence rather than an ordered pair
Nicole would have to use 400 megabytes of data for both bills to be the same of $123.
