Systems of Linear Equations

Systems of Linear Equations

In today's lesson you will learn how to answer the question, "Is (x,y) a solution to the system of equations?."

What is a system of linear equations?

*A system of linear equations is a set of two or more linear equations in the same variables.

*A system of linear equations must be solved at the same time, or simultaneously.

See the example to the right. This is an example of a system of linear equations.

Solutions of a system of linear equations:

*A solution to a system of linear equations in two variables is an ordered pair that is the solution of each equation.

In the case of our example when x = -1 and y = 2, both equations are true. The solution is (-1, 2).

Example:

Is (-1, 2) a solution to this example?

Let's find out:

1. Plug -1 in for x in both equations.
2. Plug 2 in for y in both equations.
3. Ask yourself, "Are both of the equations true?"
4. If the answer is yes, then this is a solution.

In the case of our example, (-1, 2) is a solution.

Multiple Choice

What are systems of linear equations?

• a set of two or more linear equations in the same variables
• a polynomial with three terms
• set of two or more linear inequalities in the same variables

Multiple Choice

• Yes

• No

Multiple Choice

• Yes

• No

Multiple Choice

• Yes

• No

Multiple Choice

• Yes

• No

A way of looking at things...

Another way to find out if a set of points, (x,y) is a solution is by looking at the graph.

A Graphical Approach

(One Solution)

Where ever the graphs of linear functions intersect we have a solution.

In this case we have one solution for the system y = -x + 7 and y = 2x + 1 at the point (2, 5).

A Graphical Approach

(No Solutions)

When lines never intersect, then there are no solutions to the system of equations.

Notice in the example that the lines have the same slope and different y-intercepts. (Recall: y=mx+b)

A Graphical Approach

(Infinitely Many Solutions)

Notice in the example that the equations look different at first, but after a little algebra, we see that they are the same line. There are infinitely many solutions.

Fill in the Blank

Type the solution as an ordered pair, (x, y).

Fill in the Blank

Type the solution to the system of equations as an ordered pair, (x,y).

Open Ended

Write a new equation after replacing one variable with the other variable in the problem. (Hint: replace the y's).

Multiple Choice

Write the new equation to solve after using the substitution process correctly for the system.

y = 5x - 7

-3x - 2y= -12

• -3(5x - 7) - 2y= -12

• -3x - 2(5x-7) = -12

• -3x -2y = 12(5x-7)

Multiple Choice

Write the new equation to solve after using the substitution process correctly for the system.

y = x - 1

2x - 3y= -1

• 2(x-1) - 3y= -1

• 2x - 3(x-1) = -1

• 2x -3y = 12(x-1)

Multiple Choice

Which is easier to solve for a variable?

• 1st

• 2nd

Multiple Choice

Which is the correct form of substitution?

• 3x - 8(5x-3)= - 3

• 3(5x-3)-8y = -3

• 3x - 8y = -3(5x-3)

Multiple Choice

Solve the system using the substitution method.

• (-3,0)

• (5,-3)

• (0,-3)

Multiple Choice

Which is easier to solve for an equations

• 1st

• 2nd

Multiple Choice

Which is the correct substitution process?

• -7x - 2(2y+11) = -13

• -7(2y+11) -2y = -13

• -7x - 2y = -13(2y+11)

Multiple Choice

Solve the system of equations using the substitution method.

• (4, 3)

• (3,-4)

• (3, -6)

Multiple Choice

Which equation is easier to solve for a variable.

• 1st

• 2nd

Multiple Choice

Which is the correct way to use the substitution method?

• -3x + 6(5x-2) = -12

• -3(5x-2) + 6y = -12

• -3x + 6y = -12(5x-2)

Multiple Choice

Solve the system of equations

• (-2,0)

• (0,-2)

• (0,0)

Multiple Choice

Which variable is easier to solve for in this system?

$$x+3y=15$$  

$$-8x+3y=-12$$  

• x

• y

Multiple Choice

which is the correct way to rewrite the first equation?

$$x+3y=15$$  

$$-8x+3y=-12$$  

• x=15-3y

• x=3y+15

• 3y=15+x

• y=3(x+15)

Multiple Choice

which is the correct solution to the system?

$$x+3y=15$$  

$$-8x+3y=-12$$  

• no sol'n

  parallel lines

• (6, 3)

• (4, 3)

• (3, 4)