Make-Up Tests?

More Systems of

Equations

Alt.Khan Document

February 5, 2014

February

6 7 8

11 12 13 14 15

Systems of Inequalities

Systems of Inequalities

Review Coordinate

Unit

RakeLeaves

HelpWith

Laundry

Review forSystems

Test

Test: SystemsEq./Ineq.

Begin Monomial

Unit MonomialsExponents

Exponents

No Warm-Up;Practice Problem Section of

Notebook

2014

We need to eliminate (get rid of) a variable.To simply add this time will not eliminate a variable. If there was a –2x in the 1st equation, the x’s would be eliminated when we add. So we will multiply the 1st equation by a – 2.

Like variables must be lined under each other.

x + y = 4 2x + 3y = 9

1x + 1y = 4 2x + 3y = 9

Solve Systems of Equations by Elimination (Multiplying)

2X + 3Y = 9

-2X - 2 Y = - 82X + 3Y = 9

Now add the two equations and solve.

Y = 1

THEN----

X + Y = 4( ) -2

Solve Systems of Equations by Elimination (Multiplying)

(3,1)

Substitute your answer into either original equation and solve for the second variable.

Solution

Now check our answers in both equations--

X + Y = 4

X + 1 = 4 - 1 -1

X = 3

Solve Systems of Equations by Elimination (Multiplying)

x + y = 43 + 1 = 4 4 = 4

2x + 3y = 9

2(3) + 3(1) = 9

6 + 3 = 9

9 = 9

Can you multiply either equation by an integer in

order to eliminate one of the variables?

Here, we must multiply both equations by a

(different) number in order to easily eliminate

one of the variables.

Solve Systems of Equations by Elimination (Multiplying)

Multiply the top equation by 2, and

the bottom equation by -3

EliminatePlug back in

Solve for other variable

3x – 2y = -72x -5y = 10

Write your solution as an ordered pair(-5,-4)

Plug both solutions into original equations

3x – 2y = -7-15 – (-8) = -7

-7 = - 7

2x - 5y = 10

-10 – (-20) = 10

10= 10

Solve: By Substitution

Recall that when we 'solve' a point-slope formula, we end up in slope-intercept form. In much the same way, the substitution method is closely related to the elimination method. After eliminating one variable and solving for

the other, we substitute the value of the variable back into the equation.For example: Solve 2x + 3y = -26 using elimination 4x - 3y = 2

What is the value of x ?

At this point we substitute -4 for x, and solve for y. This is exactly what the substitution method is except it is done at the beginning.

-4

Solve: By Substitution Example 1: y = 2x

4x - y = -4

4x - 2x = -4; 2x = -4; x = -2

Then, substitute -2 for x in the first equation: y = 2(-2); y = -4

Finally, plug both values in and check for equality.

-4 = 2(-2); True; 4(-2) - (-4) = -4; -8 + 4 = -4; True

Since the first equation tells us that y = 2x,

replace the y with 2x in the second equation.

3x + 2y = 8 2x – 3y = – 38

Solve: By Substitution

Solve: By Elimination

x – 2y = 73x + 2y = 13

𝒙−𝟐𝒚=𝟑𝟐 𝒙−𝟒 𝒚=𝟕

no solution

𝒙− 𝒚=𝟑

infinitely many solutions

A math test has a total of 25 problems. Some problems are worth 2 points and some are worth 3 points. The whole test is worth 63 points. How many 2-point problems were there?

x =

y =

# of 2-point problems

# of 3-point problems

25x y 2 3 63x y

2 2 2

13y

2x 2y 502 3 63x y

25x y 25x 1313 13

12x 12 2-point problems

1. Mark the text.

2. Label variables.

3. Create equations.4. Solve.

5. Check.

Applying systems of Equations:

Applying Systems of Equations:

Prom tickets cost $10 for singles and $15 for couples. Fifty more couples tickets were sold than were singles tickets. Total ticket sales were $4000.

How many of each ticket type were sold?

Applying Systems of Equations

Example 1: The sum of two numbers is 52. The larger number is 2 more than 4 times the

smaller number. Find both numbers.Example 1: x + y = 52

After reading, determine the best method to solve

x = 4y + 2

-(x + y = 52)-x - y = -52x -4y = 2 Rearrange+ _________

-5y = -50

y = 10

x + 10 = 52; x = 42

Class Work: