Guided Practice

Subtracting a positive number is the same as adding the opposite of that number. For example, 2 – 8 = 2 + (–8).

Sometimes you have to add a negative when building a linear equation, because the starting value is below the point that you are thinking about as 0. Consider the example at the right.

The lowest temperature from the video is degrees Celsius.

The highest temperature from the video is degrees Celsius.

The equation we can use to convert the temperatures is x + .

A temperature in degrees Celsius is represented by the variable .

A temperature in degrees Fahrenheit is represented by the variable .

Convert each temperature from degrees Celsius to degrees Fahrenheit.

–22.5°C = °F

45°C = °F

The rate of change for the Celsius-Fahrenheit relationship is .

So, for every increase of °C, there is an increase of °F.

When x = 0, y = .

May starts with 3 cups of water in a container. She then adds 1 cup of vinegar for every 16 cups of water to make a cleaning solution. Write an equation relating vinegar (v) and water (w) in the container.

Write a proportion: =

vw

Cross multiply: =

Add: = +

A taxi in New York City might cost you $1.60 per mile plus an initial fee of $2.50. Write an equation relating cost (c) in dollars to distance (d) in miles.

Write a proportion: =

dc

Cross multiply: =

Add: = +

In July 2008, gas cost about $4.11 per gallon. As of July 2017, this price fell by $0.20 per gallon every year. Write an equation relating the cost of gas (g) in dollars and time (t) in years, with July 2008 representing t = 0.

Write a proportion: =

tg

Cross multiply: =

Add: = +

Your favorite cereal contains 307 calories in 1 cup. Write an equation relating calories (c) to amount of cereal (a) in cups.

Write a proportion: =

ac

Cross multiply: =

Add: = +

Jasmine started out this month just $35 below her fundraising goal. But she got $3 in donations every 2 days. Write an equation to relate time (t) in days from the beginning of the month to Jasmine's amount raised (d) in dollars, relative to her goal.

Write a proportion: =

td

Cross multiply: =

Isolate, add: d = t + ()

Trinh owed his parents $28, but he paid them back using the money he earned from chores, which was $7.50 every 2 weeks. Write an equation relating the time (t) in weeks to his balance (b) in dollars.

Write a proportion: =

t2

Cross multiply: =

Isolate, add: b = t + ()

At 6 a.m. the temperature was –3°F. But the temperature increased 0.5 degree every 1.5 hours. Write an equation relating the time (t) in hours to the temperature (f) in degrees Fahrenheit, given that 6 a.m. represents t = 0.

Write a proportion: =

1.5t

Cross multiply: =

Isolate, add: f = t + ()

Jacob's family drove 142 miles every 2 hours. Write an equation relating the distance (d) in miles they drove to the time (t) in hours.

Write a proportion: =

td

Cross multiply: =

Isolate, add: d = t + ()

To solve an equation, you can use inverse operations. These are operations that "undo" each other. For example, given the expression x + 4, you can get back to x by subtracting 4.

So, x + 4 – 4 = x.

And division is the inverse of multiplication.

This number line shows that 3x ÷ 3 = x.

You can perform more than one inverse operation to solve an equation. For example, 2x + 4 = 6. What does x equal?

First, we subtract 4 from both sides of the equation to get 2x = 2. Then, we divide both sides of the equation by 2 to get x = 1.

Substitute x = 1 back into the equation to check if it is true: 2(1) + 4 = 6.

You have to watch out for this common mistake.

When you have an equation like x – 2 = 3, you can add 2 to both sides to get x by itself. But if we change the left side a little to be 2 – x = 3, then adding 2 to both sides will just give you

4 – x = 5. That's okay, but it's probably not what you wanted!

To get x by itself, you can think of 2 – x as

2 + (–x). Now you can subtract 2 from both sides to get –x = 1 and then divide both sides by –1 to get x = –1.

Watch out for division too: 2 ÷ x = 3. You can't just multiply by 2 to solve it. What should you do?

Use an inverse operation to make the equation true. Use - for subtraction and / for division.

x + 10 = x

Use an inverse operation to make the equation true. Use - for subtraction and / for division.

6x = x

Use an inverse operation to make the equation true. Use - for subtraction and / for division.

a + 7 = a

Use an inverse operation to make the equation true. Use - for subtraction and / for division.

7 • p = p

Use an inverse operation to make the equation true. Use - for subtraction and / for division.

q = q • 7

Use an inverse operation to make the equation true. Use - for subtraction and / for division.

f = f + 9

Use an inverse operation to make the equation true. Use + for addition and * for multiplication.

f = f – 9

Use an inverse operation to make the equation true. Use + for addition and * for multiplication.

m ÷ 2 = m

Solve the equation. Check your work.

2x + 6 = 10

x =

Solve the equation. Check your work.

4m + 2 = 18

m =

Solve the equation. Check your work.

9 + 3q = 24

q =

Solve the equation. Check your work.

2g ÷ 8 = 1

g =

Solve the equation. Check your work.

y/5 – 1 = 3

y =

The amount of 2D space taken up by a circle, its area, is equal to πr^{2}, where r is equal to the radius of the circle. The distance around a circle, its circumference, is equal to 2πr. The value of π is often rounded to 3.14.

The radius of a circle is the distance from the center of the circle to a point on the circle. A circle has an infinite number of radiuses (radii).

The square picture frame has side lengths of 9 in. It surrounds a circular frame.

In inches, how much more material was used to make the picture frame than the circle? (Use 3.14 for pi.)

The square frame is inches longer than the circular frame.

A goat is tied to a rope that is 4 feet long. How much grass can the goat eat? (Use 3.14 for pi.)

The goat can eat square feet of grass.

The radius of a bicycle tire is 14 inches. One full turn of the tire moves the rider forward a distance equal to the circumference of the tire. If a cyclist has traveled one-quarter mile, how many turns did the tire make? Round to the nearest hundredth. Use 3.14 for π.

The tire made about turns.

If you double the radius, the circumference of a circle increases by a factor of .

If you double the radius of a circle, the area increases by a factor of .

A water sprinkler rotates in a circle. The radius of the circle is 9 feet. What area of the lawn is watered by the sprinkler? Use 3.14 for pi.

The sprinkler waters an area of square feet.

Points

Lines

Erase

A linear equation is an equation of the form px + q = r. For example, 2x + 4 = 100 is a linear equation. You will learn how to solve the equation 2x + 4 = 100 using inverse operations. But this is not the only way to solve.

You can ask, "What number plus 4 is equal to 100?" You know that number is 96. So then you know that 2x stands for the number 96. Then ask, "What number times 2 is equal to 96?" You can then figure out that the number is 48. So, x stands for 48!

So, just remember: You don't have to follow a procedure to solve equations. Step back and THINK! ALWAYS!

Try to solve this equation: 4x + 20 = 200.

x =