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Linear Equations in Linear Algebra

A system of linear equations is two or more linear equations considered at the same time.

A solution to a system of linear equations (in two variables) is an ordered pair of real numbers
that is a solution to every equation in the system.

x + 2y = 8 is a linear equation in two variables.
Solutions to this equation include (0,4), (2,3), (4,2), (6,1), (8,0) and many, many more.

2x - y = 6 is also a linear equation in two variables.
Solutions to this equation include (0, -6), (1, -4), (2, -2), (3,0), (4,2), (5,4) and many, many more.

is a system of linear equations in two variables.

It has only one solution. What is it?

Why is (2,3) not a solution to the system

Why is (5,4) not a solution to the system

Solving Systems of Linear Equations by the Graphical Method

Example 1 Solve each system of linear equations by inspecting the graphs below it.

Solving Systems of Linear Equations by Substitution

How to use the substitution method to solve systems of linear equations:
Step 1 Solve one of the equations for one of the two variables.
Step 2 Substitute for that variable in the other equation. The result should be an
equation with just one variable.
Step 3 Solve the equation found in Step 2.
Step 4 Substitute the result from Step 3 into the equation from Step 1 to find the
value of the other variable.
Step 5 Check the solution in the equation not used in Step 1.

Example 2 Solve the system using the substitution method.

Step 1 Solve the equation x - 2y = 3 for x.

Step 2 Substitute the expression 2y + 3 for x in the equation 3x - 8y = 7.

Step 3 Solve the equation 3(2y + 3) - 8y = 7 for y.

Step 4 Since y = 1, substitute 1 for y in the equation x = 2y + 3 found in Step 1 and find x.

Step 5 Since the equation x - 2y = 3 is equivalent to the equation x = 2y + 3 just used in
Step 4, the ordered pair (5,1) should satisfy it. Check that (5,1) also satisfies the
equation 3x - 8y = 7. 3(5) - 8(1) = 7?
If so, then the solution set to the system is {(5,1)}.

Example3 Solve the system using the substitution method.

Step1 Solve the equation 2x-y=12 for y.

Step2 Since y=2x-12,substitute the expression 2x-12 for y in the equation 6x-3y=10.

Step3 Solve the equation 6x-3(2x-12)=10 for x.

Step4 Is the equation you found in Step3 a true statement or a false statement?

Step5 What is the solution set?

Example 4 Solve the system using the substitution method.

The next method we will study is called the addition or elimination method.
It goes by both names because it involves adding the two equations together
in such a way the one of the variables is eliminated from consideration.

Let’s see how this method works by using it to solve the system

What we want to do is add the two equations together in such a way that either the
x-variable or the y-variable is eliminated. If we add the left-hand side to the left-hand
side and add the right-hand side to the right-hand side, watch what happens.

The y-variable goes away because its coefficients in the two equations are additive opposites. That makes the y-terms add together to make 0y.

Once we know the value of x, we can substitute it into either equation to find the value of y.

The only solution is (3,1) which we should check in the other equation 3x + 5y = 14
just to be sure. 3(3) + 5(1) = 14, so the solution set is {(3,1)}.

Example 5 Solve the system using the elimination method.

Example 6 Solve the system using the elimination method.

Example 7 Solve each system using the elimination method.