Solve for X in 4=(2x-1)×2/3: Detailed Linear Equation Solution

Q: Why multiply by the reciprocal instead of dividing by the fraction?

Multiplying by the reciprocal is the same as dividing by a fraction, but it's clearer! When you see $ \times \frac{2}{3} $, multiply by $ \frac{3}{2} $ to cancel it out completely.

Q: What happens to the (2x-1) when I multiply by the reciprocal?

The fraction $ \frac{2}{3} $ gets canceled out, leaving you with just $ (2x-1) $! This is why multiplying by the reciprocal is so powerful - it eliminates the fraction entirely.

Q: How do I know which reciprocal to use?

For fraction $ \frac{a}{b} $, the reciprocal is always $ \frac{b}{a} $. So for $ \frac{2}{3} $, use $ \frac{3}{2} $. Just flip the numerator and denominator!

Q: Can I solve this by distributing the fraction first?

You could, but it's much messier! You'd get $ 4 = \frac{4x}{3} - \frac{2}{3} $, which is harder to work with. Always eliminate fractions first to keep calculations simple.

Q: Why does my decimal answer 3.5 match the fraction 7/2?

Because $ \frac{7}{2} = 7 \div 2 = 3.5 $! Both forms are correct. You can leave your answer as either 3.5 or $ \frac{7}{2} $ depending on what the problem asks for.

Linear Equations with Fractional Multiplication

$4=(2x-1)\times\frac{2}{3}$

How much is Xworth?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solution

00:03 Multiply by the denominator to eliminate the fraction, multiply accordingly

00:14 Divide by 2 and reduce as much as possible

00:29 Arrange the equation so that one side has only the unknown X

00:35 Isolate X

00:40 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

$4=(2x-1)\times\frac{2}{3}$

How much is Xworth?

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Eliminate the fraction by multiplying both sides of the equation by the reciprocal of $\frac{2}{3}$ .
Step 2: Simplify the resulting equation.
Step 3: Solve for $x$ .

Now, let's work through each step:

Step 1: Multiply both sides of the equation by $\frac{3}{2}$ to eliminate the fraction. This gives:

$\frac{3}{2} \times 4 = \frac{3}{2} \times \left((2x - 1) \times \frac{2}{3}\right)$

Simplifying, the left side becomes:

$6 = 2x - 1$

Step 2: Add 1 to both sides to solve for $2x$ :

$6 + 1 = 2x - 1 + 1$

$7 = 2x$

Step 3: Divide both sides by 2 to isolate $x$ :

$\frac{7}{2} = x$

Converting $\frac{7}{2}$ to a decimal gives us:

$x = 3.5$

Therefore, the solution to the problem is $x = 3.5$ .

Final Answer

$3.5$

Key Points to Remember

Essential concepts to master this topic

Fraction Elimination: Multiply both sides by reciprocal of the fraction
Technique: Multiply by $\frac{3}{2}$ to eliminate $\frac{2}{3}$
Verification: Substitute x = 3.5 back: $(2(3.5)-1) \times \frac{2}{3} = 4$ ✓

Common Mistakes

Avoid these frequent errors

Only multiplying one side by the reciprocal
Don't multiply just the right side by $\frac{3}{2}$ and leave 4 unchanged = unbalanced equation! This violates the equality principle and gives wrong solutions. Always multiply both sides by the same value to maintain balance.

Practice Quiz

Test your knowledge with interactive questions

$$ 5x=1 $$

What is the value of x?

FAQ

Everything you need to know about this question

Why multiply by the reciprocal instead of dividing by the fraction?

Multiplying by the reciprocal is the same as dividing by a fraction, but it's clearer! When you see $\times \frac{2}{3}$ , multiply by $\frac{3}{2}$ to cancel it out completely.

What happens to the (2x-1) when I multiply by the reciprocal?

The fraction $\frac{2}{3}$ gets canceled out, leaving you with just $(2x-1)$ ! This is why multiplying by the reciprocal is so powerful - it eliminates the fraction entirely.

How do I know which reciprocal to use?

For fraction $\frac{a}{b}$ , the reciprocal is always $\frac{b}{a}$ . So for $\frac{2}{3}$ , use $\frac{3}{2}$ . Just flip the numerator and denominator!

Can I solve this by distributing the fraction first?

You could, but it's much messier! You'd get $4 = \frac{4x}{3} - \frac{2}{3}$ , which is harder to work with. Always eliminate fractions first to keep calculations simple.

Why does my decimal answer 3.5 match the fraction 7/2?

Because $\frac{7}{2} = 7 \div 2 = 3.5$ ! Both forms are correct. You can leave your answer as either 3.5 or $\frac{7}{2}$ depending on what the problem asks for.

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