Question

5 describe the steps for solving the equation,
\frac{1}{5}(2x - 3) + 2 = \frac{3}{10}(5 - x) - 4x

Explanation:

Step1: Eliminate fractions

Multiply every term by 10 (the least common denominator of 5 and 10) to eliminate fractions:
$$10\times\frac{1}{5}(2x - 3)+10\times2 = 10\times\frac{3}{10}(5 - x)-10\times4x$$
Simplify each term:
$$2(2x - 3)+20 = 3(5 - x)-40x$$

Step2: Distribute terms

Distribute the coefficients outside the parentheses:
For the left side: \(2(2x - 3)=4x - 6\), so left side becomes \(4x - 6 + 20\).
For the right side: \(3(5 - x)=15 - 3x\), so right side becomes \(15 - 3x - 40x\).

Simplify both sides by combining like terms:
Left side: \(4x + 14\) (since \(-6 + 20 = 14\)).
Right side: \(15 - 43x\) (since \(-3x - 40x = -43x\)).

Step3: Isolate variable terms

Add \(43x\) to both sides to get all \(x\)-terms on the left:
$$4x + 43x + 14 = 15 - 43x + 43x$$
Simplify: \(47x + 14 = 15\).

Subtract 14 from both sides to isolate the term with \(x\):
$$47x + 14 - 14 = 15 - 14$$
Simplify: \(47x = 1\).

Step4: Solve for \(x\)

Divide both sides by 47:
$$x=\frac{1}{47}$$

Answer:

\(x = \boldsymbol{\frac{1}{47}}\)