United States of AmericaPA
High School Core Standards - Algebra I Assessment Anchors

# 5.05 Applications of linear systems

Lesson

The magic of systems of equations comes to life when we see how useful it is in real life applications. It is usually used when we have at least two unknown quantities and at least two pieces of information involving both of these quantities. The first part is to use some variables to represent these quantities and then figuring out how to write the bits of information down as equations. Then it's just a matter of using either the substitution method, elimination method or a graphical method to solve the equations simultaneously.

Let's explore further:

#### Worked examples

##### question 1

The sum of two numbers, $a$a and $b$b, is $7$7, while their product is $10$10. What are the values of $a$a and $b$b if $a$a is larger?

Think: What conditions do $a$a and $b$b need to satisfy and what possible solutions are there for each condition?

Do:

Let's take a look at $a$a & $b$b having a sum of $7$7 first. If we choose a random number like $0$0 to be $a$a, then the only possible value for $b$b is $7$7, as $0+7=7$0+7=7. Similarly if $a=1$a=1, then $b$b has to equal $6$6 as $1+6=7$1+6=7. Therefore possible number pairs include (but are not limited to):

$a$a $b$b
$0$0 $7$7
$1$1 $6$6
$2$2 $5$5
$3$3 $4$4
$4$4 $3$3
$5$5 $2$2
$6$6 $1$1
$7$7 $0$0

Now if we look at the second condition, we can choose say a value of $1$1 for $a$a, then $b$b has to be $10$10, as $1\times10=10$1×10=10. Similarly if $a$a was $2$2, then $b$b has to be $5$5 as $2\times5=10$2×5=10. Possible pairs that give us a product of $10$10 include (but are not limited to):

$a$a $b$b
$1$1 $10$10
$2$2 $5$5
$5$5 $2$2
$10$10 $1$1

We can now see that the only two pairs satisfying both conditions are $a=2$a=2 & $b=5$b=5, and $a=5$a=5 & $b=2$b=2. We know $a$a has to be larger so the answer must be $a=5$a=5 & $b=2$b=2.

##### Question 2

Leah got a quote from two photographers for an event. Photographer A charges $\$48$$48 for a booking fee plus \17$$17 per hour. Photographer B charges $\$28$$28 for a booking fee plus \21$$21 per hour.

a) Fill out a table comparing how much it would cost in total to hire each photographer for the following hours: $2,3,4,5,6$2,3,4,5,6

Think about how much it would cost to hire Photographer A for $x$x hours, make a formula and use it to see the total cost for different hours. Do the same for Photographer B.

Do

$\text{cost of hiring}=\text{booking fee}+\text{hourly rate}\times x$cost of hiring=booking fee+hourly rate×x where $x$x is the number of hours

That means the formula for

Photographer A is $\text{cost }=48+17x$cost =48+17x.

Photographer B is $\text{cost }=28+21x$cost =28+21x.

We can use these formulas to write up a table.

eg. for two hours with Photographer A, it costs $48+17\times2=\$82$48+17×2=$82.

### Outcomes

#### CC.2.2.HS.C.1

Use the concept and notation of functions to interpret and apply them in terms of their context.

#### A1.1.2.2.1

Write and/or solve a system of linear equations (including problem situations) using graphing, substitution, and/or elimination. Note: Limit systems to two linear equations.

#### A1.1.2.2.2

Interpret solutions to problems in the context of the problem situation. Note: Limit systems to two linear equations.