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Math  /  Algebra

QuestionSolve for ww. −2(−2w+5)−6w=4(w−3)−8-2(-2 w+5)-6 w=4(w-3)-8
Simplify your answer as much as possible.

Studdy Solution

STEP 1

What is this asking? We need to find the value of ww that makes this equation true! Watch out! Distributing negatives can be tricky, so let's be extra careful with those minus signs!

STEP 2

1. Distribute and Simplify
2. Combine Like Terms
3. Isolate the Variable
4. Find the Solution

STEP 3

Alright, let's **kick things off** by distributing that −2-2 on the left side of the equation.
Remember, we're multiplying −2-2 by *both* terms inside the parentheses: −2w-2w and +5+5. −2(−2w+5)=(−2)⋅(−2w)+(−2)⋅(+5)=4w−10-2(-2w + 5) = (-2) \cdot (-2w) + (-2) \cdot (+5) = 4w - 10 So, the left side becomes: 4w−10−6w4w - 10 - 6w

STEP 4

Now, let's **tackle the right side**.
We'll distribute the 44 across the (w−3)(w - 3): 4(w−3)=4⋅w+4⋅(−3)=4w−124(w - 3) = 4 \cdot w + 4 \cdot (-3) = 4w - 12 So, the right side becomes: 4w−12−84w - 12 - 8

STEP 5

After distributing on both sides, our equation now looks like this: 4w−10−6w=4w−12−84w - 10 - 6w = 4w - 12 - 8

STEP 6

On the left side, we have 4w4w and −6w-6w.
Combining these gives us: 4w−6w=−2w4w - 6w = -2w So, the left side simplifies to: −2w−10-2w - 10

STEP 7

On the right side, we have −12-12 and −8-8.
Combining these gives us: −12−8=−20-12 - 8 = -20 So, the right side simplifies to: 4w−204w - 20

STEP 8

Our equation now looks much cleaner: −2w−10=4w−20-2w - 10 = 4w - 20

STEP 9

Let's **get all the** ww **terms together** on the right side.
We can do this by adding 2w2w to *both* sides of the equation: −2w−10+2w=4w−20+2w-2w - 10 + 2w = 4w - 20 + 2w −10=6w−20-10 = 6w - 20

STEP 10

Now, let's **move the constants** to the left side by adding 2020 to both sides: −10+20=6w−20+20-10 + 20 = 6w - 20 + 20 10=6w10 = 6w

STEP 11

Finally, to **solve for** ww, we'll divide both sides by 66: 106=6w6\frac{10}{6} = \frac{6w}{6} 106=w\frac{10}{6} = w

STEP 12

We can **simplify** 106\frac{10}{6} by dividing both the numerator and denominator by their greatest common factor, which is 22: 106=10÷26÷2=53\frac{10}{6} = \frac{10 \div 2}{6 \div 2} = \frac{5}{3} So, w=53w = \frac{5}{3}.

STEP 13

w=53w = \frac{5}{3}

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