Systems of Equations Review

Systems of Equations Review

What is a system of equtions?

A system of linear equations is a collection of two or more iinear equations.

Recall that the graph of a line contains all ordered pairs (x, y) that satisfy its equation.

For example, the graph of the line

y = 3x - 2 contains all ordered pairs (x, y) that make the equation true (see picture at right).

Similarly, the graph of the line

y = -x + 6 contains all ordered pairs (x, y) that make the equation true (see picture at right).

Multiple Choice

If we graph both lines together, what will be the solution(s) of the system?

• The ordered pair (x,y) of the point where the two lines intersect.

• The ordered pair (x, y) of any point where the two lines do not intersect.

• The system will never have a solution.

• Any point will be a solution.

The solution of the system will be where the two lines intersect.

The intersection point is on both lines, and its ordered pair makes both equations true.

Look at the graphs of y = 3x - 2 and

y = -x + 6 together.

What is the solution of this system?

Summary:

To solve a system by graphing, find all points where the lines intersect.

Multiple Choice

True or False?

A system of equations could have no solutions.

• True

• False

Answer: True

If the lines are parallel, they never intersect, and the equation has no solutions.

Multiple Choice

True or False?

A system of equations could have exactly two solutions.

• True

• False

Answer: False

Two lines cannot possibly intersect in only two different places.

Multiple Choice

True or False?

A system of equations could have infinitely many solutions.

• True

• False

Answer: True

This one is tricky - a system could have infintely many solutions if the two equations look different, but describe the same line.

For example, y = 2x + 3 and

2y = 4x + 6 are the same line, since you can divide 2 from both sides of the second equation to get the first.

The lines "intersect" at every point, so any point along the line is a solution.

There are infinitely many solutions.

NOTE: The ininitely many solutions are all along the line. This does not mean that every possible ordered pair is a solution.

(For example, (2, 1) is not a solution in our picture.)

Recall that we can also solve systems algebraically (and this is how we usually do it!)

Multiple Choice

When solving systems algebraically, what happens if there are no solutions?

• You will reach a true statement (e.g. 5 = 5) during the solution process.

• You will reach 0 = 0 during the solution process

• You will reach an untrue statement (e.g., 1 = 5) during the solution process.

• You will not be able to start the solution process.

Multiple Choice

When solving systems algebraically, how do you know if there are infinitely many solutions?

• This only happens if the equations look identical from the start.

• You will reach an untrue statement (e.g., 1 = 5) during the solution process

• You will not be able to start the solution process.

• You will reach a true statement (e.g. 5 = 5) during the solution process.

Now, let's review:

Solving Systems by Substitution

Example:

4x + 2y = 8

3x + 5y = 13

Please write down work for this example on paper.

Step 1: Choose one equation and solve for a variable.

You can choose either equation and either variable.

4x + 2y = 8

3x + 5y = 13

Let's solve for y in the first equation:

4x + 2y = 8

2y = -4x + 8

y = -2x + 4

Step 2: Substitute the expression for the variable into the other equation.

We had y = -2x + 4.

Substitute into Equation 2 (3x + 5y = 13):

3x + 5(-2x + 4) = 13

3x -10x + 20 = 13

-7x + 20 = 13

-7x = -7

x = 1

Step 3: After finding the value of one variable, substitute it into an equation and solve for the other variable.

You can substiute the value into either of the original equations.

We had x = 1.

Let's substitute x = 1 into Equation 1:

4(1) + 2y = 8

4 + 2y = 8

2y = 4

y = 2

We have x = 1 and y = 2.

Thus, for our system

4x + 2y = 8

3x + 5y = 13

Solution: (1, 2)

Now, let's review:

Solving Systems by Elimination

Example:

4x - 2y = 6

3x - 4y = -8

Please write down work for this example on paper.

Step 1: Multiply both sides of one or both equations by a constant so that one variable is eliminated when the equations are added together.

Sometimes, you only need to do this to one equation.

Other times, you may need to do it to both equations.

You can eliminate either variable, but one may be easier.

There may be multiple ways to eliminate the variable that you choose.

Let's try to eliminate y.

4x - 2y = 6 (multiply by -2)

3x - 4y = -8

Note: If you were eliminating x, you would need to multiply both sides of both equations (e.g., multiply Equation 1 by -3 and Equation 2 by 4).

Step 2: Add the two equations together to eliminate one variable and solve the new equation.

You will be left with an equation with one variable to solve.

-8x + 4y = -12

3x - 4y = -8

__________

-5x = -20

x = 4

Step 3: After finding the value of one variable, substitute it into an equation and solve for the other variable.

You can substiute the value into either of the original equations.

We had x = 4.

Let's substitute x = 4 into Equation 1:

4(4) - 2y = 6

16 - 2y = 6

-2y = -10

y = 5

We have x = 4 and y = 5.

Thus, for our system

4x - 2y = 6

3x - 4y = -8

Solution: (4, 5)