february 11, 2015,

Begin Unit On Systems of

Equations/Inequalities

Practice:

a. Together

b. On your own

Khan Academy Topics Posted & Due

by the 17th.:

Today:

I Happened to notice

this last night.....

Systems of Equations To solve for one variable, only one equation is needed. When solving for two variables, two or more equations

are required to know what the solutions may be. Three unknowns require three equations, ....

These are systems of equations.

SYSTEMS OF LINEAR EQUATIONS

So far, we have solved equations with one variable: 3x + 5 = 35and found combinations of solutions in two variables.

Now we will solve multiple equations at the same time, looking for an ordered pair which solves each equation, and thus is a solution for both.

Example:

3x + 3y = -3y = x + 1

We’ll take a quick look at all of the methods before focusing on each one separately.

3x + 5y = 35

There are 3 methods for solving systems of equations: 1) By Graphing 2) By Elimination 3) By Substitution

Example:

3x + 3y = -3y = x + 1

SOLVING SYSTEMS BY GRAPHING:

x

y

1. Write each equation into slope-intercept form.

2. Graph both equations in the same coordinate plane.

3. Find the point of intersection.

4. Check your answer.- Plug that point into both equations and make sure that it is true for both.

y mx b

Example:

3x + 3y = -3y = x + 1

SOLVING SYSTEMS BY ELIMINATION:

1. Arrange the like variables in columns.

2. Pick a variable, x or y, and make the two equations opposites using multiplication.

3. Add the equations together (eliminating a variable) and solve for the remaining variable.

4. Substitute the answer into one of the ORIGINAL equations and solve.

5. Check your solution.

SOLVING SYSTEMS BY SUBSTITUTION:

Example:

3x + 3y = -3y = x + 1

1. Solve one of the equations for x or y.

2. Substitute your new expression from Step 1 into the other equation and solve for the variable.

3. Plug that solved variable into the other equation from Step 1 and solve for the other variable.

4. Check your answers by plugging it into the original equations.

- Get x or y by itself.

Solving Linear Systems by graphing

x

y

Consider the following system: x – y = –1

x + 2y = 5

Using the graph to the right, we can see that any of these ordered pairs will make the first equation true since they lie on the line.

Notice: Any of these points will make the second equation true.

However, there is ONE point that makes both true together…

(1, 2)

The point where they intersect makes both equations true at the same time, and is the solution to this system

Graph this line

Then this line

Plug the coordinates into both equations to check if each equations is true.

SOLUTIONS TO SYSTEMS OF LINEAR EQUATIONS

If the system of linear equations is going to have a solution, then the solution will be an ordered pair (x , y) where x and ymake both equations true at the same time.

These are the three possible solutions:

Practice: 1 of 3

x

y

Rewrite the two equations in slope-

intercept form:

Plot points for each line.

Draw the lines.

These two equations represent the same line.

Therefore, this system of equations has infinitely many solutions.

Solve by Graphing

The two equations in slope-intercept

form are:

x

y

Plot points for each line.

Draw in the lines.This system of equations represents two parallel lines.

This system of equations has no solution because these two lines have no points in common.

Practice: 2 of 3

x

y

The two equations in slope-intercept form are:

Plot points for each line.Draw in the lines.

This system of equations represents two intersecting lines.

The solution to this system of equations is a single point (3,0)

Practice: 3 of 3

SOLVING SYSTEMS BY ELIMINATION:(3)

1. Arrange the like variables in columns.

2. Pick a variable, x or y, and make the two equations opposites using multiplication.

3. Add the equations together (eliminating a variable) and solve for the remaining variable.

4. Substitute the answer into one of the ORIGINAL equations and solve.

5. Check your solution.

Tomorrow....

a. Solving by Substitution

Right Now......

Class Work 3.3

Rewrite the inequality 4x < -𝟐

𝟑y +

𝟏

𝟐in standard form

with integer values only.24x + 4y < + 3

Inequalities

4x + 𝟏

𝟑y < -

𝟏

𝟐

5 min. break, then the last test of the coordinate plane unit.

Solving Systems of Equations

Tomorrow:

V.3 # 2

V.4 # 4; E. None