Solve for X: Finding the Solution to -1/2x + 3 = 7x - 27

Q: Why multiply by 2 instead of just working with the fraction?

Multiplying by 2 eliminates the fraction completely, making the equation much easier to solve! Working with $ -x + 6 = 14x - 54 $ is simpler than dealing with $ -\frac{1}{2}x $ throughout.

Q: How do I know which number to multiply by to clear fractions?

Look at the denominator of each fraction. In this case, we only have $ \frac{1}{2} $, so multiply by 2. If you had multiple fractions, use the LCD (Least Common Denominator).

Q: Can I check my answer a different way?

Yes! You can also graph both sides as separate equations: $ y = -\frac{1}{2}x + 3 $ and $ y = 7x - 27 $. They intersect at x = 4.

Q: What if I get a negative answer?

Negative answers are totally normal! Don't assume you made a mistake just because x is negative. Always verify by substituting back into the original equation.

Linear Equations with Fractional Coefficients

Solve for X:

$-\frac{1}{2}x+3=7x-27$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:04 We want to isolate the unknown X

00:21 Let's reduce what we can

00:32 Collect terms

00:39 Isolate the unknown X

00:55 Let's reduce what we can

01:13 Convert from mixed number to improper fraction

01:16 Multiply by the reciprocal fraction to isolate X

01:29 Let's reduce what we can

01:38 Make sure to multiply numerator by numerator and denominator by denominator

01:41 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve for X:

$-\frac{1}{2}x+3=7x-27$

Step-by-step solution

To solve the equation $-\frac{1}{2}x + 3 = 7x - 27$ , follow these steps:

Step 1: Eliminate the fraction
Multiply every term in the equation by 2 to remove the fraction: $2 \times \left(-\frac{1}{2}x + 3\right) = 2 \times (7x - 27)$ This simplifies to: $-x + 6 = 14x - 54$
Step 2: Move variable terms to one side
Add $x$ to both sides to move the $x$ -terms to one side: $-x + x + 6 = 14x + x - 54$ This simplifies to: $6 = 15x - 54$
Step 3: Move constant terms to the other side
Add 54 to both sides to move the constant terms: $6 + 54 = 15x - 54 + 54$ $60 = 15x$
Step 4: Solve for $x$
Divide both sides by 15 to isolate $x$ : $\frac{60}{15} = x$ , $x = 4$

Therefore, the solution to the equation is $x = 4$ .

Final Answer

$4$

Key Points to Remember

Essential concepts to master this topic

Rule: Multiply both sides by same number to eliminate fractions
Technique: Multiply by 2 to clear $-\frac{1}{2}x$ becoming $-x$
Check: Substitute x = 4: $-\frac{1}{2}(4) + 3 = -2 + 3 = 1$ and $7(4) - 27 = 1$ ✓

Common Mistakes

Avoid these frequent errors

Only multiplying the fractional term by 2
Don't multiply just $-\frac{1}{2}x$ by 2 and leave other terms unchanged = unbalanced equation giving wrong answer! This breaks the equality because you're not doing the same operation to both sides. Always multiply every single term on both sides by the same number.

Practice Quiz

Test your knowledge with interactive questions

$$ x+7=14 $$

$x=\text{?}$

FAQ

Everything you need to know about this question

Why multiply by 2 instead of just working with the fraction?

Multiplying by 2 eliminates the fraction completely, making the equation much easier to solve! Working with $-x + 6 = 14x - 54$ is simpler than dealing with $-\frac{1}{2}x$ throughout.

What if I move variables to the right side instead?

That works too! You could subtract 14x from both sides to get $-15x + 6 = -54$ , then solve. The final answer will be the same either way.

How do I know which number to multiply by to clear fractions?

Look at the denominator of each fraction. In this case, we only have $\frac{1}{2}$ , so multiply by 2. If you had multiple fractions, use the LCD (Least Common Denominator).

Can I check my answer a different way?

Yes! You can also graph both sides as separate equations: $y = -\frac{1}{2}x + 3$ and $y = 7x - 27$ . They intersect at x = 4.

What if I get a negative answer?

Negative answers are totally normal! Don't assume you made a mistake just because x is negative. Always verify by substituting back into the original equation.

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