Worksheet for solving multi-step equations and applying algebra to geometry.

Let's solve the problems step by step from your worksheet titled "Solving Multi-Step Equations Worksheet".

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Problem 1: Solve each of the following equations

We'll solve each equation one at a time.

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#### a) $ 3x + 4 = 2x $

Step 1: Subtract $ 2x $ from both sides to get all $ x $-terms on one side:
$$
3x - 2x + 4 = 0 \Rightarrow x + 4 = 0
$$

Step 2: Subtract 4 from both sides:
$$
x = -4
$$

✔ Answer: $ x = -4 $

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#### b) $ 6 + 2x = x $

Step 1: Subtract $ x $ from both sides:
$$
6 + 2x - x = 0 \Rightarrow 6 + x = 0
$$

Step 2: Subtract 6 from both sides:
$$
x = -6
$$

✔ Answer: $ x = -6 $

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#### c) $ 4x - 7 = 3x $

Step 1: Subtract $ 3x $ from both sides:
$$
4x - 3x - 7 = 0 \Rightarrow x - 7 = 0
$$

Step 2: Add 7 to both sides:
$$
x = 7
$$

✔ Answer: $ x = 7 $

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#### d) $ 7 - 7x = 2x - 3 $

Step 1: Move all $ x $-terms to one side and constants to the other.
Add $ 7x $ to both sides:
$$
7 = 9x - 3
$$

Step 2: Add 3 to both sides:
$$
10 = 9x
$$

Step 3: Divide both sides by 9:
$$
x = \frac{10}{9}
$$

✔ Answer: $ x = \frac{10}{9} $

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#### e) $ -7 - 6x = -6x + 19 $

Step 1: Add $ 6x $ to both sides:
$$
-7 = 19
$$

✘ This is not true — contradiction!

This means no solution exists (the equation is inconsistent).

✔ Answer: No solution

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#### f) $ 50 + 23x = 8x + 95 $

Step 1: Subtract $ 8x $ from both sides:
$$
50 + 15x = 95
$$

Step 2: Subtract 50 from both sides:
$$
15x = 45
$$

Step 3: Divide by 15:
$$
x = 3
$$

✔ Answer: $ x = 3 $

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Problem 2: Check your solutions (do a ± 0.5 check) for questions 2a and 2b above

Wait — this seems like a typo in the worksheet. It says "questions 2a and 2b", but we only have questions labeled a) through f) in Problem 1.

Looking at the image, it appears that Problem 2 refers to checking equations a) and b) from Problem 1.

So let’s assume:

- 2a) refers to equation a): $ 3x + 4 = 2x $
- 2b) refers to equation b): $ 6 + 2x = x $

And we are to check our solutions using a ±0.5 check, meaning plug in values slightly above and below the solution to see if they satisfy the equation.

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#### 2a) Check $ x = -4 $ in $ 3x + 4 = 2x $

Try $ x = -4 $:
- Left: $ 3(-4) + 4 = -12 + 4 = -8 $
- Right: $ 2(-4) = -8 $
→ Equal ✔

Now try $ x = -3.5 $ (within ±0.5):
- Left: $ 3(-3.5) + 4 = -10.5 + 4 = -6.5 $
- Right: $ 2(-3.5) = -7 $
→ Not equal → not a solution

Try $ x = -4.5 $:
- Left: $ 3(-4.5) + 4 = -13.5 + 4 = -9.5 $
- Right: $ 2(-4.5) = -9 $
→ Not equal

Only $ x = -4 $ works. So solution is correct.

✔ Check passed

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#### 2b) Check $ x = -6 $ in $ 6 + 2x = x $

Try $ x = -6 $:
- Left: $ 6 + 2(-6) = 6 - 12 = -6 $
- Right: $ -6 $
→ Equal ✔

Now try $ x = -5.5 $:
- Left: $ 6 + 2(-5.5) = 6 - 11 = -5 $
- Right: $ -5.5 $
→ Not equal

Try $ x = -6.5 $:
- Left: $ 6 + 2(-6.5) = 6 - 13 = -7 $
- Right: $ -6.5 $
→ Not equal

Only $ x = -6 $ satisfies.

✔ Check passed

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Problem 3: The perimeter of the triangle is 17 cm. Determine the length of each side by setting up an equation and solving for the unknown.

From the diagram (as described):

- One side is $ x $
- Another is $ x + 3 $
- Third is $ x + 4 $

Perimeter = sum of all sides:
$$
x + (x + 3) + (x + 4) = 17
$$

Simplify:
$$
x + x + 3 + x + 4 = 17 \Rightarrow 3x + 7 = 17
$$

Subtract 7:
$$
3x = 10
$$

Divide by 3:
$$
x = \frac{10}{3} \approx 3.33\,\text{cm}
$$

Now find each side:

- Side 1: $ x = \frac{10}{3} \approx 3.33 $ cm
- Side 2: $ x + 3 = \frac{10}{3} + 3 = \frac{10}{3} + \frac{9}{3} = \frac{19}{3} \approx 6.33 $ cm
- Side 3: $ x + 4 = \frac{10}{3} + 4 = \frac{10}{3} + \frac{12}{3} = \frac{22}{3} \approx 7.33 $ cm

Check perimeter:
$$
\frac{10}{3} + \frac{19}{3} + \frac{22}{3} = \frac{51}{3} = 17 \quad ✔
$$

✔ All correct

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✔ Final Answers:

#### Problem 1:
a) $ x = -4 $
b) $ x = -6 $
c) $ x = 7 $
d) $ x = \frac{10}{9} $
e) No solution
f) $ x = 3 $

#### Problem 2:
- a) $ x = -4 $: Checked, valid
- b) $ x = -6 $: Checked, valid

#### Problem 3:
- Side lengths:
$ x = \frac{10}{3} $ cm ≈ 3.33 cm
$ x+3 = \frac{19}{3} $ cm ≈ 6.33 cm
$ x+4 = \frac{22}{3} $ cm ≈ 7.33 cm
- Perimeter: $ 17 $ cm ✔

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Let me know if you'd like this formatted as a printable answer sheet!