Activity Set 3.7PREP PPTX

Visual Algebra for Teachers

Chapter 3

REAL NUMBERS AND QUADRATIC FUNCTIONS

Visual Algebra for Teachers

Visual Algebra for Teachers

Activity Set 3.7

Systems of Equations

PURPOSETo learn: To use the substitution and elimination methods of

solving systems of equations.

To understand how these solutions can be viewed graphically

Graphing calculator with table functions

MATERIALS

INTRODUCTION

SYSTEMS OF EQUATIONS A system of equations is two or more equations of the same basic type.

A general rule for solving a system of equations is that you need the same number of equations as “unknowns” or variables. That is, you can solve a system of two equations in two unknowns, a system of three equations in three unknowns, etc. There are two main techniques for solving a system of equations: The Substitution Method and the Elimination Method.

The Substitution Method This is a system of two equations in two unknowns

1) Solve for one variable in one equation

2 3 6

4 4 7

m n

m n

6 32 3 6 2 6 3

2Solve for (this is just one way, solving for also works)

nm n m n m

m n

The Substitution Method

2) Substitute the resulting value into the other equation to reduce the number of variables in the second equation.

2 3 6

4 4 7

m n

m n

6 3

2

nm

6 34 4 7 4 4 7

2

6 3Substitute into second equation

2

m

nm n n

nm

The Substitution Method Simplify and then solve the new equation for the second variable (n):

192 3 6

2

4 3 19 12

4 12 57 45

45

4

m

m

m

m

6 34 4 7

2

2 6 3 4 7

12 2 7

192 19

2

nn

n n

n

n n

Substitute the value for n back into either equation and solve for m.

The Substitution Method The final solution to this system is

Check your solutions:

These values create valid equations.

2 3 6

45 192 3 6

4 2

2 45 6 19 24

90 114 24

m n

19 45

2 4n m

4 4 7

45 194 4 7

4 2

4 45 8 19 28

180 152 28

m n

The Elimination Method Multiply one or more equations by a carefully chosen constant and then add the result.

For example, if we multiply the first equation by -2, we can “eliminate” m when we add the equations:

1. System

2 3 6

4 4 7

m n

m n

2. Multiply by -2

2 2 3 2 6

4 4 7

m n

m n

3. Simplify

4 6 12

4 4 7

m n

m n

The Elimination Method

Just like with Substitution; substitute the value for n back into either equation and solve for m.

19 45

2 4n m

5. Simplify

19

2n

4. Add

4 6 12

4 4 7

0 2 19

m n

m n

n

Question #1

Jack and Jill and Little Boy Blue have competing lemonade stands on Rosie Lane. For c cups of lemonade and p dollars of profit, the relationship between c and p can be described using the following equations

For Jack and Jill : 10 20

For Little Boy Blue : 5 25

c p

c p

Question #1aWrite in your coursepack, Activity Set 3.7 #1a

If Jack and Jill sell 30 cups of lemonade, what is their profit? (Hint: Use the J & J equation)

If Little Boy Blue sells 30 cups of lemonade, what is his profit? (Hint: Use the LBB equation)

Question #1bWrite in your coursepack, Activity Set 3.7 #1b

If Jack and Jill make a profit of $5.00, how many cups of lemonade did they sell? (Hint: Use the J & J equation)

If Little Boy Blue makes a profit of $5.00, how many cups of lemonade did he sell? (Hint: Use the LBB equation)

Question #1cdeWrite in your coursepack, Activity Set 3.7 #1cde

c. For the lemonade stands, profit is a function of the cups of lemonade sold. What type of function is this?

d. Solve each equation for p to determine p as a function of c. What type of function is this? Is this the same answer you had in the previous part?

e. For 0 cups of lemonade, how much profit do Jack and Jill make? Little Boy Blue? What do these values, the p-intercepts, mean in this context?

Question #1fgWrite in your coursepack, Activity Set 3.7 #1fg

f. By looking at the function equations, will the functions intersect? How can you tell by just looking at the function equations?

g. Graph the two functions and mark any points of intersection on the graph. Scale your axes carefully and label everything clearly. Do negative values for c make any sense here?

Question #1hWrite in your coursepack, Activity Set 3.7 #1h

Use the Substitution Method to solve the original system of equations and determine where Jack and Jill and Little Boy Blue make the same amount of profit.

Give your answers in a sentence including the correct units (cups, dollars).

Question #1iWrite in your coursepack, Activity Set 3.7 #1i

Use the Elimination Method to solve the original system of equations and determine where Jack and Jill and Little Boy Blue make the same amount of profit.

Give your answers in a sentence including the correct units (cups, dollars).

You are now ready for:

PREP QUIZ 3.7

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Visual Algebra for Teachers