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Math  /  Algebra

Questionf) 2x3−x12=−2\frac{2 x}{3}-\frac{x}{12}=-2

Studdy Solution

STEP 1

What is this asking? We need to find the value of xx that makes this equation true! Watch out! Fractions can be tricky!
Make sure to find a common denominator before combining them.

STEP 2

1. Prepare the equation
2. Isolate xx
3. Verify the solution

STEP 3

We've got two fractions with different denominators: 3 and 12.
To combine them, we need a common denominator.
The smallest number that both 3 and 12 divide into is **12**.
Let's rewrite the first fraction so it has a denominator of 12.
We can do this by multiplying the top and bottom by **4**:
2x3⋅44=8x12\frac{2x}{3} \cdot \frac{4}{4} = \frac{8x}{12}

STEP 4

Now, let's put this back into our original equation:
8x12−x12=−2\frac{8x}{12} - \frac{x}{12} = -2Much better!
Now we can work with those fractions!

STEP 5

Since the fractions now have the same denominator, we can subtract the numerators:
8x−x12=7x12=−2\frac{8x - x}{12} = \frac{7x}{12} = -2So, our equation becomes:
7x12=−2\frac{7x}{12} = -2

STEP 6

To get xx by itself, we need to get rid of that fraction.
We can do this by multiplying both sides of the equation by **12**:
7x12⋅12=−2⋅12\frac{7x}{12} \cdot 12 = -2 \cdot 127x12⋅121=−24\frac{7x}{12} \cdot \frac{12}{1} = -24Dividing by twelve and multiplying by twelve add to one, so they disappear, and we get:
7x=−247x = -24

STEP 7

Almost there!
Now, we just need to divide both sides by **7** to isolate xx:
7x7=−247\frac{7x}{7} = \frac{-24}{7}Dividing by seven and multiplying by seven add to one, so they disappear, and we get:
x=−247x = -\frac{24}{7}Boom! We found xx!

STEP 8

Let's plug our value of xx back into the original equation to make sure it works:
2⋅(−247)3−−24712=−2\frac{2 \cdot (-\frac{24}{7})}{3} - \frac{-\frac{24}{7}}{12} = -2

STEP 9

−4873+24712=−2\frac{-\frac{48}{7}}{3} + \frac{\frac{24}{7}}{12} = -2
−487⋅13+247⋅112=−2-\frac{48}{7} \cdot \frac{1}{3} + \frac{24}{7} \cdot \frac{1}{12} = -2−4821+2484=−2-\frac{48}{21} + \frac{24}{84} = -2−167+27=−2-\frac{16}{7} + \frac{2}{7} = -2−147=−2-\frac{14}{7} = -2−2=−2-2 = -2It works!

STEP 10

x=−247x = -\frac{24}{7}

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