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Chapter 7: Problem 105

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{3 x+7}{3 x+10}=\frac{8}{11}$$

Short Answer

Expert verified
The given statement is true, as the solution x = 1/3 satisfies original equation.

Step by step solution

01

Cross Multiplication

Cross multiply the fractions in the equation: \(3x + 7\) * 11 = \(3x + 10\) * 8.
02

Simplify Both Sides

After cross multiplication, simplify both sides of the equation to get 33x + 77 = 24x + 80.
03

Rearrange the Equation

Re-arrange this equation, bringing like terms together to balance the equation. By subtracting 24x from both sides, we get: 9x + 77 = 80. By subtracting 77 from both sides, we get: 9x = 3.
04

Solve for x

Now, divide both sides of the equation by 9 to isolate x. The solution is: x = 3/9 = 1/3.
05

Verifying Solution

Substitute the solution x = 1/3 into the original equation to check for balance: (3*1/3 + 7) / (3*1/3 + 10) = 8/11.
06

Prove the statement

After substituting \(x = 1/3\) in the original equation, it gives: 8/11 = 8/11. Therefore the statement is indeed true.

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