Solving Systems of Linear Equations by Elimination and Substitution

A system of linear equations is a set of two or more linear equations.
We can solve a system of equations algebraically (by elimination or substitution) or .

Solving a system of equations algebraically is like solving a riddle. Each linear equation provides a clue needed to solve for the mystery values of each variable.

Just like the answer to a mystery must agree with each clue, the values of each variable in a system of equations must agree with each equation in the system.

This means that once we find the values that make every equation in the system true at the same time, we have our solution.

Consider the following system of equations:

Our riddle is: If
, what are
equal to?

We can solve the systems of equations by :

Eliminate and solve for :

Plug in and solve for using one of the equations from the system:
Mystery solved!

We typically see systems of linear equations with one solution, but they can also have no solutions or infinite solutions depending on if and how the lines of the equations intersect.

How do we solve a system of linear equations by graphing?

To solve a system of linear equations using the graphing method, we simply need to graph each equation in the system and find the point where all the lines intersect.

For example, we can see that the solution to the following system of equations is :

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How to Solve a System of Linear Equations Algebraically

Solving a System of Linear Equations by

Step 1. Line up the variables and constants.

Rearrange the terms so that the variables and constants in each equation line up.

Step 2. Select a variable to eliminate.

If possible, pick a variable that has coefficients that have the same absolute value or are multiples of each other.

Step 3. Match the coefficients of the variable that is getting eliminated.

Multiply each equation by the right factor to make the coefficient of the variable match in both equations.

Step 4. Eliminate the variable.

Add or subtract the two equations so that the variable cancels out.

Step 5. Solve for the remaining variable.

Step 6. Plug in the value of the known variable into either of the equations to solve for the eliminated variable.

The Elimination Method
The Elimination Method
The Substitution Method
The Substitution Method
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