Module 11.1

Solving Systems Of Linear Equations

How do you find the solution of a system of linear equations by graphing?

P. 479

𝟐𝒙 + 𝒚 = 𝟔 is a linear equation.It has an infinite number of solutions; meaning there are an infinitely many numbers that can be substituted for x and y, in order that they satisfy the equation.For example, here are just a few:

Here is a graph of this equation.

y-intercept

x-intercept

P. 480

−𝒙 + 𝒚 = 𝟑 is also a linear equation.It too has an infinite number of solutions.For example, here are just a few:

Here is a graph of this equation.

y-intercept

x-intercept

When we say a System of Linear Equations, we mean two or more linear equations.The two previous examples are 𝟐𝒙 + 𝒚 = 𝟔 and −𝒙 + 𝒚 = 𝟑Here they are, graphed individually, then combined:

𝟐𝒙 + 𝒚 = 𝟔 −𝒙 + 𝒚 = 𝟑 Combined

All points on the graphs of each equation represent the solutions to those equations.

If any point is on both graphs, then it is a solution of both equations. In other words,a solution of a System of Linear Equations is any ordered pair that satisfies all of theequations in the system. It does that at thepoint where the lines intersect each other.

And since they intersect at only one point, there is only one solution: The ordered pair appears to be (1 , 4).

What we’re doing here is calledSolving Linear Systems By Graphing.

𝟐𝒙 + 𝒚 = 𝟔−𝒙 + 𝒚 = 𝟑

Yes, it appeared to be (1 , 4).But is that really correct?

You can determine that by using Algebra.Substitute 𝒙 = 𝟏 and 𝒚 = 𝟒 into bothequations, and seeing if those two numberssatisfy both.

−𝒙 + 𝒚 = 𝟑−𝟏 + 𝟒 = 𝟑 True

𝟐𝒙 + 𝒚 = 𝟔𝟐 𝟏 + 𝟒 = 𝟔 True

𝟐𝒙 + 𝒚 = 𝟔−𝒙 + 𝒚 = 𝟑

They do!

Slope =

Y-Intercept =

Plot the 1st point at ________

Rise =Run

Plot the 2nd point at ________

Find the x and yintercepts

What is theIntersecting point?

__________

Check it by substituting.

P. 481

P. 485

P. 488

Wren and Jenni are reading the same book. Wren is on page 12 and reads 3 pages every night. Jenni is on page 7 and reads 4 pages every night. After how many nights will they have read the same number of pages? _________

How many pages will that be? ________

Wren = 𝑦 = 12 + 3𝑥 Jenni = 𝑦 = 7 + 4𝑥

𝑦 = 3𝑥 + 12 𝑦 = 4𝑥 + 7

Solve by graphing. Give an approximate solution if necessary.

0 1 2 3 4 5 6 7 8 9

Number Of Nights

50

40

30

20

10

0

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P. 485

There are two types of Special Linear Systems.Here’s the first.

P. 482

Here’s Special Linear System # 2 P. 482

We’ve just done Solving Linear Systems By Graphing.

Coming up…

11.2 will be Solving Linear Systems By Substitution.11.3 will be Solving Linear Systems By Adding or Subtracting.11.4 will be Solving Linear Systems By Multiplying First.

12.1 will be Creating Systems Of Linear Equations.12.2 will be Graphing Systems Of Linear Inequalities.