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Chapter 0: Problem 32

Solve the equation and check your solution. (If not possible, explain why.) $$ \frac{10 x+3}{5 x+6}=\frac{1}{2} $$

Short Answer

Expert verified
The solution to the equation \(\frac{10x+3}{5x+6}=\frac{1}{2}\) is \(x = 0\).

Step by step solution

01

Cross-multiplication

Cross-multiplication is the method to remove the fractions from the equation. By applying this, we get: \( (10x+3) * 2 = (5x + 6) * 1\). After multiplication, simplify to: \(20x + 6 = 5x + 6\).
02

Simplification

Bring like terms together in the equation, which means to subtract \(5x\) from both sides of the equation. The equation simplifies to: \(20x - 5x = 5x - 5x + 6 - 6\). Further simplify to: \(15x = 0\).
03

Solve for x

To isolate x, divide both sides by 15. The final equation is: \(x = 0 / 15\). Simplifying further gives: \(x = 0\).
04

Check The Solution

Substitute \(x = 0\) back into the original equation to check if the left-hand side equals the right-hand side. \( \frac{10*0+3}{5*0+6} = \frac{1}{2} \). This simplifies to \( \frac{1}{2} = \frac{1}{2} \). Since these are equal, the solution is correct.

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