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Solve equations containing rational coefficients

Solving equations with rational coefficients involves interpreting fractions and terminating decimals as coefficients in one-variable linear relationships and applying equality-preserving transformations to isolate the variable; multiplying every term by a common nonzero denominator clears fractions without changing the solution set. The understanding includes verifying solutions by substitution and recognizing when a linear equation has one solution, no solution, or infinitely many solutions, without extending to nonlinear equations or systems.

Detailed Explanation: Solve equations containing rational coefficients

Rational coefficients can be fractions or terminating decimals. To solve an equation with fractions, multiply every term by the least common denominator. This clears the fractions while keeping both sides equal.

Consider:

12x+34=52\frac{1}{2}x+\frac{3}{4}=\frac{5}{2}

Step 1: Find a common denominator.

The denominators are 22 and 44, so the least common denominator is 44.

Step 2: Multiply every term by 44.

4(12x)+4(34)=4(52)4\left(\frac{1}{2}x\right)+4\left(\frac{3}{4}\right)=4\left(\frac{5}{2}\right)

Simplify:

2x+3=102x+3=10

Step 3: Isolate the variable.

Subtract 33 from both sides:

2x=72x=7

Divide both sides by 22:

x=72x=\frac{7}{2}

Step 4: Check the solution.

Substitute 72\frac{7}{2} for xx in the original equation:

12(72)+34=74+34=104=52\frac{1}{2}\left(\frac{7}{2}\right)+\frac{3}{4} =\frac{7}{4}+\frac{3}{4} =\frac{10}{4} =\frac{5}{2}

Both sides are equal, so the solution is

x=72\boxed{x=\frac{7}{2}}

Learn by doing: Solve equations containing rational coefficients

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    Linear Equation - One Variable, Three Terms

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