How to Solve Simultaneous Equations – Quiz
In this post, we look at how to solve simultaneous equations problems on your algebra test.
Instructions: Use the elimination method to solve systems of equations questions 1 to 3. Use the substitution method to solve systems of equations questions 4 and 5.
[WpProQuiz 32]Solve Simultaneous Equations – Linear Equations
For simultaneous equations problems, you will see two equations that have common variables, such as x or y.
You will see systems of equations questions where both of the equations are linear equations.
Elimination Method Example
Look the example below, in which both equations are linear.
Example: Solve for a and b if a + b = 3 and 3a – b = 17.
Answer: a = 5 and b = –2
We will use the elimination method in the solution that follows.
STEP 1: Make one equation
Make one equation from the two equations as shown.
You make one equation by combining the left sides of the two equations together, then placing the equals sign in the new equation, and then combining the right sides together.
a + b = 3 and 3a – b = 17 →
a + b + 3a – b = 3 + 17
STEP 2: Group like terms together
Group like terms together and simplify.
a + b + 3a – b = 3 + 17
a + 3a + b – b = 3 + 17
a + 3a – b + b = 3 + 17
a + 3a = 3 + 17
4a = 20
STEP 3: Divide to solve
Perform division to solve for a.
4a = 20
4a ÷ 4 = 20 ÷ 4
a = 5
STEP 4: Replace variable with its value
Replace variable “a” with its value in one of the equations.
a = 5 and a + b = 3 so,
a + b = 3 → 5 + b = 3
STEP 5: Solve for the other variable
Solve for the other variable, which is “b” in this problem.
5 + b = 3
5 – 5 + b = 3 – 5
5 – 5 + b = 3 – 5
b = 3 – 5
b = –2
The example above assumes that your systems of equations have like terms.
If this is not the case, you will need to perform multiplication in order to create like terms before you carry out the elimination method.
Click on the following link for help with advanced problems on the elimination method.
Solve Systems of Equations – Linear & Quadratic Equations
In the example below, one equation is linear and the other is quadratic.
Example: Solve for x and y if x2 + y2 = 9 and y – 3x = 1
STEP 1: Isolate variable y in the linear equation
In order to solve systems of equations in this type of question, you need to isolate y in the linear equation:
y – 3x = 1
y – 3x + 3x = 1 + 3x
y – 3x + 3x = 1 + 3x
y = 1 + 3x
STEP 2: Substitute 1 + 3x for y
In the quadratic equation, you need to substitute 1 + 3x for y.
x2 + y2 = 9 is our quadratic equation.
x2 + y2 = 9
(1 + 3x)2 + y2 = 9
STEP 3: FOIL Method and Quadratic Formula
You then need to use the FOIL method and the quadratic formula to solve systems of equations problems that include quadratic equations.
Please see our exercises on the FOIL method and the quadratic formula for further exercises on these skills.
You can also solve systems of equations problems graphically.
We will cover this skill in our posts on graphing.
More Exam Preparation
If you are taking a math placement test, you may also want to view the following:
How to Factor in Algebra Problems
