| {{ 'ml-lesson-number-slides' | message : article.intro.bblockCount }} |
| {{ 'ml-lesson-number-exercises' | message : article.intro.exerciseCount }} |
| {{ 'ml-lesson-time-estimation' | message }} |
Here are a few recommended readings before getting started with this lesson.
Zain and Jordan planned an adventure across a vast desert landscape. They will cycle for three days until they reach a huge music festival! The first two days, Zain and Jordan need to cycle the same distance each day. On the third day, they will have 10 miles remaining to be cycled.
The total distance Zain and Jordan cycle during the trip is 60 miles. If they miscalculate their trip, they will miss the festival!
Equations can be named according to the minimum number of inverse operations needed to solve them.
LHS+ 3=RHS+ 3
Add terms
.LHS / 2.=.RHS / 2.
a* b/c=a/c* b
Calculate quotient
Identity Property of Multiplication
Zain and Jordan are cycling along. Already on their first day, they ran into a problem! Zain's tire got a terrible flat and they do not have a spare to replace it. They need to buy a new tire.
They head into town to buy extra tires to be better prepared.
.LHS /3.=.RHS /3.
Cross out common factors
Cancel out common factors
Calculate quotient
t= 17
Multiply
Subtract term
undoeach other. Consider the given equation. Here, the variable m is divided by 2 and then 5 is added to the result.
LHS * 2=RHS* 2
a/2* 2 = a
Multiply
m= 30
Calculate quotient
Add terms
Inverse operations and the Properties of Equality are used to move all the variable terms to one side of the equation. In this case, the Subtraction Property of Equality is used to subtract x from both sides of the equation. 3x &= x-2 &⇓ 3x - x &= x -2 - x
Then, the equation is simplified by combining like terms. This results in an equation with just one variable term. 3x - x &= x -2 - x &⇓ 2x &= - 2
On the second day of this ride to the festival, Zain and Jordan begin to wonder about some of the data from their ride.
LHS * 5/-4=RHS* 5/-4
Commutative Property of Multiplication
a/b* b/a=1
a*b/c= a* b/c
Multiply
- a/- b=a/b
Calculate quotient
t= 25
Calculate quotient
Subtract term
LHS-x=RHS-x
Subtract terms
LHS * (-1)=RHS* (-1)
- a(- b)=a* b
a/c* b = a* b/c
LHS * 3=RHS* 3
a/3* 3 = a
Multiply
x= 30
a/c* b = a* b/c
Multiply
Calculate quotient
Subtract term
Solve the equations by using the Properties of Equality. If necessary, give answers as decimals rounded to two decimal places.
Zain and Jordan continued cruising along their cycling trip. The beauty of the ride became even more noticeable as they could hear songbirds! Some birds sang perched atop a power line.
A few more birds flew in and joined the flock. As a result, the number of birds doubled. Then, 4 birds flew away. In the end, 8 birds were left singing on the power line.
Original Number of Birds: b The number of birds on the power line after more birds flew in is twice the original number of birds, or 2b. After 4 birds flew away, the number of birds on the power line becomes 2b-4. This expression is equal to 8. Using this information, an equation can be written. 2b-4 = 8 This equation models the given situation.
.LHS /2.=.RHS /2.
Cross out common factors
Cancel out common factors
Calculate quotient
Zain and Jordan successfully reached the music festival! The line to enter the festival is super long. Each came up with their own way to describe the difference between the time they expected to wait in line and the time they will actually have to wait.
Expected Wait Time: t Zain's and Jordan's comments suggest two ways to write the actual wait time in terms of t. Zain said that the actual wait time was three times the expected wait time, or 3t. According to Jordan, the actual wait time was 40 minutes longer than expected, or t + 40 minutes. Actual Wait Time in Zain's Words:& 3t Actual Wait Time in Jordan's Words:& t+40 In both cases, the value of the expression is the same. In other words, the actual wait time of Zain and Jordan is equal. This allows an equation to be written by setting both person's expressions equal to each other. 3t = t + 40
.LHS /2.=.RHS /2.
Cross out common factors
Cancel out common factors
Calculate quotient
t= 20
Multiply
An equation that models the challenge presented at the beginning of this lesson can now be written and solved. Recall that Zain and Jordan planned to cycle the same distance for the first two days of their trip, and then cycle the remaining 10 miles on the last day.
Remember, the whole trip is 60 miles long. Making these calculations will ensure they make it to the festival on time!
Number of Miles: d Next, the total number of miles can be expressed in terms of d. On each of the first two days, Zain and Jordan covered d miles. That means they cycled 2d miles the first two days. On the third day, they cycled 10 miles. The total number of miles cycled equals the first two days 2d plus the final 10 miles. Total Number of Miles: 2d+10 Finally, this expression is set to equal the total length of the trip, or 60 miles. 2d+10=60
2d+10=60
First, d is multiplied by 2 and then 10 is added to the product. These operations are undone in reverse order using inverse operations. The first operation to be undone is the addition. This is done using the Subtraction Property of Inequality..LHS /2.=.RHS /2.
Cross out common factors
Simplify quotient
Calculate quotient