How do you verify that an ordered pair solves a linear equation with fractions?

Conditions

• The candidate solution values must already be known.
• The equation being checked contains those variables with fractional coefficients.

Reasoning, step by step

1. Substitute the known values of $x$ and $y$ into the left-hand side of the equation.
2. Multiply each fraction by its corresponding integer value (multiply the numerator, keep the denominator).
3. Identify that the resulting fractions have different denominators and cannot be combined directly.
4. Find the least common multiple of the denominators to use as a common denominator.
5. Rewrite each fraction as an equivalent fraction with the common denominator by multiplying numerator and denominator by the appropriate factor (a form of 1).
6. Perform the addition or subtraction on the numerators while keeping the common denominator.
7. Reduce the resulting fraction to lowest terms by dividing numerator and denominator by their greatest common factor.
8. Compare the simplified result to the right-hand side of the original equation. If they are equal, the solution is verified.

Example

To check $(2, -2)$ in $\frac{3}{4}x + \frac{1}{5}y = \frac{11}{10}$: substitute to get $\frac{3}{4}(2) + \frac{1}{5}(-2) = \frac{6}{4} - \frac{2}{5}$. The common denominator of 4 and 5 is 20. Rewrite: $\frac{6}{4} \cdot \frac{5}{5} = \frac{30}{20}$ and $\frac{2}{5} \cdot \frac{4}{4} = \frac{8}{20}$. Subtract: $\frac{30}{20} - \frac{8}{20} = \frac{22}{20}$. Reduce: $\frac{22}{20} = \frac{11}{10}$. Since $\frac{11}{10} = \frac{11}{10}$, the solution is correct.

Common misconceptions

• Attempting to subtract fractions with unlike denominators directly without first finding a common denominator.
• Believing that rewriting a fraction with a different denominator (e.g., $\frac{6}{4}$ to $\frac{30}{20}$) changes its value.
• Forgetting to reduce the final fraction to lowest terms before comparing it to the right-hand side.

Watch the explanation

Explore next

• Why must you find a common denominator before subtracting $6/4$ and $2/5$?
• Why does multiplying by $5/5$ or $4/4$ keep a fraction equivalent?
• How does $22/20$ simplify to $11/10$ in this check?