Solve for X: Finding X in (1/4)(x-16) + (3/4)x = 8x-11

Q: Why multiply the whole equation by 4 instead of working with fractions?

Multiplying by the LCD eliminates fractions and makes the equation much easier to solve! Working with whole numbers like 4x - 16 = 32x - 44 is simpler than dealing with $ \frac{1}{4} $ and $ \frac{3}{4} $ throughout.

Q: What happens if I don't multiply every single term by 4?

You'll create an unbalanced equation that's no longer true! Every term on both sides must be multiplied by the same number to maintain equality. Missing even one term will give you the wrong answer.

Q: How do I combine like terms when I have coefficients and constants?

Group x terms together and constants together separately. In this problem: x - 16 + 3x becomes (x + 3x) + (-16) = 4x - 16.

Q: Can I check my answer by plugging x = 1 back into the simplified equation?

It's better to check using the original equation with fractions! This catches any errors you might have made during simplification. Substitute x = 1 into $ \frac{1}{4}(1-16)+\frac{3}{4}(1)=8(1)-11 $.

Linear Equations with Mixed Fractions

Solve for X:

$\frac{1}{4}(x-16)+\frac{3}{4}x=8x-11$

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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

00:07 Open parentheses properly, multiply by each factor

00:29 Negative times positive is always negative

00:42 Collect like terms

00:55 Calculate the fraction

01:05 Isolate the unknown X

01:30 Simplify what's possible, collect like terms

01:46 Isolate the unknown X

01:56 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve for X:

$\frac{1}{4}(x-16)+\frac{3}{4}x=8x-11$

Step-by-step solution

To solve this problem, we'll eliminate fractions by multiplying the entire equation by the least common denominator, and then proceed with algebraic manipulation:

Step 1: Multiply every term by the least common denominator, $4$ , to eliminate fractions:
$4 \times \left(\frac{1}{4}(x - 16) + \frac{3}{4}x\right) = 4 \times (8x - 11)$ .
Step 2: This simplifies to: $1(x - 16) + 3x = 32x - 44$ .
Step 3: Distribute within the brackets:
$x - 16 + 3x = 32x - 44$ .
Step 4: Combine like terms on the left-hand side:
$4x - 16 = 32x - 44$ .
Step 5: To isolate terms containing $x$ , subtract $4x$ from both sides:
$-16 = 28x - 44$ .
Step 6: Add 44 to both sides to further isolate $x$ :
$28 = 28x$ .
Step 7: Divide both sides by 28 to solve for $x$ :
$x = 1$ .

Therefore, the solution to the problem is $x = 1$ .

Final Answer

$1$

Key Points to Remember

Essential concepts to master this topic

Fraction Elimination: Multiply every term by LCD to clear all fractions
Distribution: Apply $\frac{1}{4}$ to both x and -16 inside parentheses
Verification: Substitute x = 1 back into original equation to confirm both sides equal ✓

Common Mistakes

Avoid these frequent errors

Forgetting to distribute the fraction to all terms in parentheses
Don't just multiply $\frac{1}{4}$ by x and ignore the -16 = wrong simplification! This skips part of the equation and leads to incorrect answers. Always distribute fractions to every term inside parentheses: $\frac{1}{4}(x-16) = \frac{1}{4}x - \frac{1}{4} \cdot 16$ .

Practice Quiz

Test your knowledge with interactive questions

$$ x+7=14 $$

$x=\text{?}$

FAQ

Everything you need to know about this question

Why multiply the whole equation by 4 instead of working with fractions?

Multiplying by the LCD eliminates fractions and makes the equation much easier to solve! Working with whole numbers like 4x - 16 = 32x - 44 is simpler than dealing with $\frac{1}{4}$ and $\frac{3}{4}$ throughout.

What happens if I don't multiply every single term by 4?

You'll create an unbalanced equation that's no longer true! Every term on both sides must be multiplied by the same number to maintain equality. Missing even one term will give you the wrong answer.

How do I combine like terms when I have coefficients and constants?

Group x terms together and constants together separately. In this problem: x - 16 + 3x becomes (x + 3x) + (-16) = 4x - 16.

Can I check my answer by plugging x = 1 back into the simplified equation?

It's better to check using the original equation with fractions! This catches any errors you might have made during simplification. Substitute x = 1 into $\frac{1}{4}(1-16)+\frac{3}{4}(1)=8(1)-11$ .

What if my LCD calculation gives me a different number?

For denominators 4 and 4, the LCD is definitely 4. If you're getting confused, list the multiples: 4 is the smallest number that both 4 and 4 divide into evenly.

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