Multi-Step Equations
3p+8=35
1st Subtract 8 from both sides (Subtraction POE) → 3p = 27
2nd Divide by 3 on both sides (Division POE) → p=9
Answer: p=9
If you add the same number to both sides of an equation, which property of equality are you using?
Addition Property of Equality
5x=20
How many solutions does this equation have?
One Solution and x = 4
Julie is 3 years older than her sister. Julie's age is 11. How old is Julie's sister?
s + 3 = 11 (Julie is 3 years older than her sister)
s = 8
Her sister is 8 years old
Solve for y:
y+k=p
Subtract k on both sides (Subtraction POE)
Answer: y=p−k
Solve the multi-step equation:
3(x−4)+2x=28
1st Distribute → 3x−12+2x=28
2nd Combine like terms → 5x−12=28
3rd Add 12 to both sides (Addition POE) → 5x=40
4th Divide by 5 on both sides (Division POE) → x=8
Answer: x=8
Solve: If 3x + 7 = 22, which property of equality would you use first to isolate x?
Subtraction Property of Equality
If solving an equation leads to a false statement like 10=31, how many solutions does it have?
No Solutions because 10 does not equal 31.
We are at a Seattle Seahawks game. We buy 4 hotdogs and 1 drink. We spend a total of $17. If the drink costs $5, how much does one hotdog cost?
4h + 5 = 17
4h = 12
h = 3 (One hotdog costs $3)
Solve for x:
ax+b=c
1st subtract b on both sides → ax = c - b
2nd divide a on both sides → x = c-b/a
Answer: x = c-b/a
4x+27=-x+42
1st Add x to both sides (Addition POE)
→ 4x + 27 + x = 42
2nd Combine like terms → 5x + 27 = 42
3rd Subtract 27 on both sides (Subtraction POE)
→ 5x = 15
4th Divide by 5 on both sides (Division POE) x=3
Answer: x=3
True or False: The Multiplication Property of Equality states that you can multiply by different numbers on both sides of an equation and the equation will still be true.
False
True or False: If two sides of an equation are identical after simplifying, (like 8=8 or 5x-2=5x-2), the equation has one solution.
False — it has infinitely many solutions because we can choose any number for x and the equation will stay equal.
Justice is shopping for her first iPad. She wants to spend a total of no more than $1,500 but knows she needs to spend at least $800 to get one that meets her needs. There is a 5% sales tax on the purchase price, and she also has to pay a$50 setup fee.
Justice wants to find the price of an iPad that fits her budget, considering a 5% sales tax and a $50 setup fee.
Write and solve a compound inequality and determine which correctly describes the price of the laptop she should consider?
ANSWER: $714.29 ≤ p ≤ $1,380.95
$800 ≤ 1.05p + 50 ≤ $1,500
First break the inequality into 2:
$800 ≤ 1.05p + 50 and 1.05p + 50 ≤ $1,500
800 ≤ 1.05p + 50 1.05p + 50 ≤ 1500
750 ≤ 1.05p 1.05p ≤ 1450
p ≥ 714.29 p ≤ 1,380.95
Solve for a:
P=2a+2b
1st subtract 2b on both sides (Subtraction POE)
P - 2b = 2a
2nd divide by 2 on both sides (Division POE)
P - 2b / 2 = a
Answer: a = P−2b / 2
Solve 7−2(3−x)=x+9
1st Use Distribute Property → 7 - 6 + 2x = x + 9
2nd Combine Like Terms → 1 + 2x = x + 9
3rd Subtract x on both sides (Subtraction POE) → 1 + x = 9
4th Subtract 1 on both sides → x = 8
Answer: x = 8
A teacher writes: 5x+2=17. Explain step by step which properties of equality you would use to solve for x.
Step 1: Subtraction POE (subtract 2 on both sides)
Step 2: Division POE (divide 5 on both sides)
A student solves and ends up with x=7 and says it only has one solution, why?
The equation only has one solution because it is only one value that makes the equation true, 7.
Milani bought b books that each cost c dollars. Milani paid a flat shipping fee of s dollars. The total cost of the books and shipping was t dollars.
This can be represented by the equation bc + s = t. Write an equation that shows the cost of each book, c.
1st Subtract s on both sides (Subtraction POE)
→ bc = t - s
2nd Divide b on both sides (Division POE)
→ c = t-s/b
Solve for x:
m = p−qx / r
1st multiply by r on both sides (Multiplication POE)
rm = p - qx
2nd subtract p on both sides (Subtraction POE)
rm - p = -qx
3rd divide by -q on both sides (Division POE)
Answer: -rm/q = x
Solve the equation:
4x−7=2(x+5)+x
1st Use Distribute Property → 4x−7=2x+10 + x
2nd Combine Like Terms → 4x−7=3x+10
3rd Subtract 3x on both sides (Subtraction POE) → x−7=10
4th Add 7 on both sides (Addition POE) → x=17
Answer: x=17
A student writes: 5(x+2)=25 ⇒ 5x+10 = 25. Which property did they use, and what would you do next?
Distributive Property ⇒ Next use Subtraction POE
7x+2=7x+10
How many solutions does this equation have?
No solution. Because when we simplify the equation, we get 2 = 10 and 2 does not equal 10.
Drea wants to make a mix of almonds, peanuts, and raisins. She wants 1 part almond, 2 parts peanuts, and 5 parts raisins. Almonds cost $7 dollars per pound, peanuts cost $4 dollars per pound, and raisins cost $2 dollars per pound. Drea has $25 to spend.
How much trail mix can Drea afford to make?
Drea can make one pound of trail mix.
almonds → 7x
peanuts → 2(4x)
raisins → 5(2x)
7x + 2(4x) + 5(2x) = 25
Distributive Property → 7x + 8x + 10x = 25
Combine Like Terms → 25x = 25
Division POE → x = 1
Solve for y:
2y + 3x = 12
1st subtract 3x on both sides (Subtraction POE)
2y = 12 - 3x
2nd divide by 2 on both sides (Division POE)
y = 12 - 3x / 2