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Math  /  Algebra

QuestionSolve the equation for xx. 310x−45=−2\frac{3}{10} x-\frac{4}{5}=-2

Studdy Solution

STEP 1

What is this asking? We need to find the value of xx that makes the equation 310x−45=−2\frac{3}{10} x-\frac{4}{5}=-2 true. Watch out! Fractions can be tricky!
Don't forget about those common denominators!

STEP 2

1. Prepare the equation
2. Isolate the term with xx
3. Solve for xx

STEP 3

Alright, let's **rewrite** our equation: 310x−45=−2 \frac{3}{10} x-\frac{4}{5}=-2 We want to work with a common denominator to make things easier.
The **least common denominator** for 10 and 5 is **10**!

STEP 4

Let's **convert** that 45\frac{4}{5} by multiplying the numerator and denominator by **2**: 45⋅22=4⋅25⋅2=810 \frac{4}{5} \cdot \frac{2}{2} = \frac{4 \cdot 2}{5 \cdot 2} = \frac{8}{10} Now, our equation looks like this: 310x−810=−2 \frac{3}{10} x-\frac{8}{10}=-2 Much better!

STEP 5

To **isolate** the term with xx, we need to **add** 810\frac{8}{10} to both sides of the equation.
Remember, what we do to one side, we *must* do to the other! 310x−810+810=−2+810 \frac{3}{10} x-\frac{8}{10} + \frac{8}{10}=-2 + \frac{8}{10} 310x=−2+810 \frac{3}{10} x = -2 + \frac{8}{10}

STEP 6

Let's **convert** that -2 into a fraction with a denominator of 10.
We can write -2 as −21\frac{-2}{1} and multiply the numerator and denominator by **10**: −21⋅1010=−2⋅101⋅10=−2010 \frac{-2}{1} \cdot \frac{10}{10} = \frac{-2 \cdot 10}{1 \cdot 10} = \frac{-20}{10} So, our equation becomes: 310x=−2010+810 \frac{3}{10} x = \frac{-20}{10} + \frac{8}{10}

STEP 7

Now, we can **add** those fractions on the right side: 310x=−20+810 \frac{3}{10} x = \frac{-20 + 8}{10} 310x=−1210 \frac{3}{10} x = \frac{-12}{10}

STEP 8

To **solve** for xx, we need to **multiply** both sides of the equation by 103\frac{10}{3}.
This will isolate xx by turning the coefficient of xx into 1. 103⋅310x=103⋅−1210 \frac{10}{3} \cdot \frac{3}{10} x = \frac{10}{3} \cdot \frac{-12}{10}

STEP 9

Now, let's **simplify**.
On the left side, the 10's divide to one and the 3's divide to one, leaving just xx.
On the right side, the 10's divide to one. x=−123 x = \frac{-12}{3}

STEP 10

Finally, we can **divide** -12 by 3: x=−4 x = -4 Boom! We found it!

STEP 11

x=−4x = -4

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