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Chapter 1: Problem 59

Solve each equation, if possible. $$ \frac{7}{3 x+10}=\frac{2}{x-3} $$

Short Answer

Expert verified
x = 41

Step by step solution

01

Cross-Multiply the Fractions

To solve the equation \(\frac{7}{3x + 10}=\frac{2}{x - 3}\), begin by cross-multiplying to eliminate the fractions. Multiply each numerator by the opposite fraction's denominator: \[7(x - 3) = 2(3x + 10)\]
02

Distribute Multiplication

Expand each side of the equation by distributing the multiplication: \[7x - 21 = 6x + 20\]
03

Move Variable Terms to One Side

Rearrange the equation to bring all terms involving \(x\) to one side. Subtract \(6x\) from both sides: \[7x - 6x - 21 = 20\] \[x - 21 = 20\]
04

Isolate the Variable

Add 21 to both sides to isolate \(x\): \[x = 20 + 21\] \[x = 41\]
05

Check for Extraneous Solutions

Verify that the solution \(x = 41\) does not make any denominators zero. For \(3x + 10\): \[3(41) + 10 = 123 + 10 = 133 eq 0\]For \(x - 3\): \[41 - 3 = 38 eq 0\] Since neither denominator is zero, the solution is valid.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

cross-multiplication
Cross-multiplication is a key step in solving rational equations. It involves multiplying the numerator of one fraction by the denominator of the other. This method helps eliminate the fractions and convert the equation into a simpler form.
For example, in the equation \(\frac{7}{3x + 10}=\frac{2}{x - 3}\), we cross-multiply to get \[7(x - 3) = 2(3x + 10).\]
Cross-multiplying is useful for dealing with equations that involve two fractions set equal to each other. This step reduces the complexity and makes it easier to isolate the variable later.
solving rational equations
Rational equations are equations that contain fractions with polynomials in the numerator and/or the denominator. Solving these equations typically involves several steps:
  • Cross-multiplication to eliminate the fractions.
  • Applying the distributive property to expand expressions.
  • Rearranging terms to isolate the variable.
  • Checking for extraneous solutions.

For our specific problem, after cross-multiplication, we apply these steps systematically to find the solution. Always remember to verify if the found solution makes any denominator zero as it can invalidate the solution.
variable isolation
Variable isolation is the process of rearranging an equation to get the variable by itself on one side. This often involves moving terms to different sides of the equation and simplifying.
In our exercise, after distributing the multiplication, we have \[7x - 21 = 6x + 20.\]
We then move all terms involving \(x\) to one side by subtracting \(6x\) from both sides:
\[7x - 6x - 21 = 20.\]
Simplifying this gives us \[x - 21 = 20.\]
Finally, we add 21 to both sides to isolate \(x\), resulting in \[x = 41.\]
This step-by-step process ensures that \(x\) is fully isolated and its correct value is found.
distributive property
The distributive property is a fundamental concept in algebra that allows you to multiply a single term by each term in a parenthesis. It is especially useful when solving equations.
For our example, after cross-multiplying, we have \[7(x - 3) = 2(3x + 10).\]
Applying the distributive property, we get:
\[7x - 21 = 6x + 20.\]
This step is critical as it transforms the equation into a form where you can more easily combine like terms and isolate the variable. By distributing the multiplication, you simplify the equation, making it possible to isolate and solve for the variable.

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