Problem 48 - Elementary and Intermediate Algebra

Question

Solve each proportion. $$ \frac{2}{q}=\frac{q-3}{2} $$

Step-by-Step Solution

Answer

There is no solution to the equation as it results in a contradiction.

Step 1: Cross-multiply the Proportion

Multiply both sides of the equation by the denominators to eliminate the fractions. This means multiplying \(2\) by \(2\), and \(q\) by \(q-3\). The equation becomes:\[ 2(q-3) = q \cdot 2 \] which simplifies to \[ 2q - 6 = 2q \]

Step 2: Subtract \(2q\) from Both Sides

Subtract \(2q\) from both sides of the equation to isolate the constant terms. This gives us:\[ 2q - 6 - 2q = 2q - 2q \]Simplifying, we have:\[ -6 = 0 \]

Step 3: Verify Equation Possibility

Since \(-6 = 0\) is not a true statement, this indicates that there is no solution to the equation because we've eliminated the variable \(q\) and are left with a false statement.

Key Concepts

Cross-Multiplication

Cross-multiplication is a straightforward technique used when dealing with proportions. When you have an equation set up as a fraction equal to another fraction, like \( \frac{2}{q} = \frac{q-3}{2} \), you can eliminate the fractions by cross-multiplying. This involves multiplying the numerator of one fraction by the denominator of the other fraction, and vice versa.
- For instance, in our exercise, you multiply 2 from the left side by \( q-3 \) on the right side.- Simultaneously, you multiply \( q \) from the right side by 2 on the left.
Thus, you get the equation \( 2(q-3) = q \cdot 2 \). This cross-multiplication transitions a proportion into an equation without fractions, making it easier to solve.
Cross-multiplication is a handy tool whenever you need to clear the fractions in an equation, and it's especially useful for simplifying algebraic equations.

Algebraic Equations

Algebraic equations are mathematical statements that include variables and constants. Solving algebraic equations, like when you solve for \( q \) in the equation\( 2q - 6 = 2q \), typically involves performing operations to isolate the variable.
- First, simplify both sides if possible. In this case, there is nothing to simplify on either side until you perform operations.- Next, perform operations like addition, subtraction, multiplication, or division to get all terms including the variable on one side and the rest on the other.
In our solved equation, after performing cross-multiplication, you subtract \( 2q \) from both sides to attempt to isolate \( q \). This yields \( -6 = 0 \), which leads us to another important concept, as the variable is canceled out.

No Solution

Sometimes when solving algebraic equations, you might end up with a statement that makes no sense, such as \( -6 = 0 \). This means there is no solution to the equation you started with. Here's why:
- After simplifying and attempting to isolate the variable, if the variable disappears and you're left with a false statement, no value of the variable will satisfy the original equation.- In this specific case, eliminating \( 2q \) from both sides leaves us with an untrue statement, indicating that the original proportion \( \frac{2}{q} = \frac{q-3}{2} \) has no solution within the set of real numbers.
"No solution" is the conclusion when a problem resolves to a contradiction. This can occur due to the structure of the equation or improper initial conditions, and it's critical to verify your final statements when solving such problems.