Problem 29

Question

Solve the equation (if possible). $$\frac{5 x-4}{5 x+4}=\frac{2}{3}$$

Step-by-Step Solution

Verified

Answer

The solution to the equation is $ x = 4 $.

1Step 1: Cross Multiply

In order to solve the equation, initially cross multiply, which is multiplying the denominator of the first fraction by the numerator of the second fraction and the denominator of the second fraction by the numerator of the first one. Doing so yields: $ (5x-4)*3=(5x+4)*2 $.

2Step 2: Simplify Each Side

Now, distribute the values on both sides of the equation to simplify: This results in $ 15x - 12 = 10x + 8 $.

3Step 3: Group like terms

Group the like terms together, moving all 'x' terms to the left side and the numeric terms to the right. This involves subtracting 10x from both sides and adding 12 to both sides which results in $ 15x - 10x = 8 + 12 $.

4Step 4: Solve for x

Simplify the developed equation, which reduces it to $ 5x = 20 $ and then divide by 5 on both sides, which gives $ x = 4 $.

Key Concepts

Cross MultiplicationSimplifying ExpressionsSolving Linear Equations

Cross Multiplication

Cross multiplication is a useful method when solving rational equations, especially when you have two fractions set equal to each other. In the equation $ \frac{5x - 4}{5x + 4} = \frac{2}{3} $, cross multiplication helps eliminate the fractions, making the equation easier to manage.Here's how it works:

Take the numerator of the first fraction $ (5x - 4) $ and multiply it by the denominator of the second fraction $ 3 $.
Then, take the numerator of the second fraction $ 2 $ and multiply it by the denominator of the first fraction $ (5x + 4) $.

This gives us the equation: \[(5x - 4) * 3 = (5x + 4) * 2\]By performing cross multiplication, you convert the equation from a rational equation into a linear format which is simpler to solve. This is often the first step in clearing fractions and simplifying equations.

Simplifying Expressions

After cross multiplication, you'll often need to simplify expressions on both sides of the equation. Simplifying involves distributing multiplication over addition or subtraction and combining like terms.From our equation:\[15x - 12 = 10x + 8\]Here's what we do:

Distribute $3$ to $5x$ and $-4$ resulting in $15x - 12$.
Similarly, distribute $2$ to $5x$ and $4$ resulting in $10x + 8$.

Once the multiplication is completed, check both sides of the equation for like terms. Combining them simplifies your work, keeping your calculations clear and manageable. Be careful with signs during distribution to ensure accuracy.

Solving Linear Equations

Once simplified, the rational equation becomes a linear equation, which is straightforward to solve. The key steps involve isolating variable terms on one side and constant terms on the other.For our example:

Start with $15x - 10x = 8 + 12$.
Subtract $10x$ from $15x$, which results in $5x$.
Add $12$ to $8$, resulting in $20$.

You end up with the simple equation:\[5x = 20\]To find the value of $x$, divide both sides by $5$:\[x = \frac{20}{5} = 4\]Solving linear equations requires balancing actions to keep the equation equal, using operations that simplify until you isolate $x$. This method works effectively, especially after initial simplifications have already reduced complexity.

Problem 29

Problem 30

Other exercises in this chapter

Problem 29

Solve the equation algebraically. Then write the equation in the form $f(x)=0$ and use a graphing utility to verify the algebraic solution. $$12(x+2)=15(x-4)-

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Problem 29

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Problem 30

Solve the inequality and sketch the solution on the real number line. Use a graphing utility to verify your solution graphically. $$0 \leq \frac{x+3}{2}

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Problem 30

Find all solutions of the equation algebraically. Check your solutions. $$\sqrt{x+5}=\sqrt{2 x-5}$$

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