Problem 31 - Algebra 1

Question

Solve the proportion. Check for extraneous solutions. $$\frac{-2}{a-7}=\frac{a}{5}$$

Step-by-Step Solution

Answer

The solutions to the proportion are a = 2 and a = 5, with no extraneous solutions.

Step 1: Apply Cross Multiplication

To solve the proportion, cross multiply. Cross-multiplication is the process of canceling out denominators in order to solve the equation. The equation becomes -2 * 5 = a * (a - 7), which simplifies to -10 = a^2 - 7a.

Step 2: Arrange the Equation into Quadratic Form

Next, rearrange the equation into the standard quadratic form, \(ax^2 + bx + c = 0\). In doing so, the equation becomes a^2 - 7a + 10 = 0.

Step 3: Solve Quadratic Equation

Next, solve the quadratic equation. The quadratic formula is used here which is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substituting the values into the quadratic formula gives the solutions for 'a' as a = 5 and a = 2.

Step 4: Check for Extraneous Solutions

Lastly, after finding the solutions, substitute them back into the original equation to check for any extraneous solutions. Upon substitution, it's determined that both a = 5 and a = 2 are valid solutions, as neither cause the denominator in the original equation to equal zero. Thus, there are no extraneous solutions in this case.

Key Concepts

Understanding Quadratic Equations

Quadratic equations are fundamental in algebra, featuring a unique structure that includes a squared variable. Each quadratic has the form \(ax^2 + bx + c = 0\). Here, 'a', 'b', and 'c' are constants with 'a' not equal to zero. This equation is integral when solving problems involving quadratic functions.

• The term \(ax^2\) is the quadratic term, involving the variable raised to the second power.
• The \(bx\) term represents the linear component, directly proportional to the variable.
• The constant term 'c' stands alone without accompanying variables.

When solving quadratic equations, methods include factoring, using the quadratic formula, or completing the square. In our exercise, the quadratic formula solution was chosen, highlighted by its ability to find the root accurately even in more complex scenarios.One crucial aspect is the discriminant \(b^2 - 4ac\), which helps determine the nature of roots. If positive, expect two real and distinct roots. If zero, there is exactly one real root. While negative implies no real roots, indicating complex solutions.

Exploring Cross Multiplication

Cross multiplication is a key technique for solving equations involving proportions. It essentially bypasses fractions by cross-referencing denominators and numerators. This approach stems from the property that if \(\frac{a}{b} = \frac{c}{d}\) then \(a \cdot d = b \cdot c\).

• Cross multiplication provides a quick method to eliminate fractions, simplifying calculations.
• It is highly effective in setting equations in forms easier for further manipulation, like turning them into quadratics.
• In the given exercise, performing cross multiplication resulted in \(-10 = a^2 - 7a\), redirecting the problem into a quadratic equation framework.

Remember, this tool only aids when an equation equates two fractions. Always ensure conditions for cross multiplication are met: balanced equations.

Investigating Extraneous Solutions

Extraneous solutions often surface when solving rational equations, although they seem valid mathematically, they are not actual solutions to the initial equation.

• Such solutions can emerge because the steps involved in reaching a solution, like squaring both sides, might introduce possibilities not present in the original context.
• Particularly in proportion problems, substituting found solutions back into the original form ensures they do not make any denominator zero.
• In the solved exercise \(\frac{-2}{a-7} = \frac{a}{5}\), checking revealed no extraneous solutions, since neither solution for 'a' led to a zero in the original denominator \(a-7\).

Regularly reverting and cross-verifying with the initial values is best practice to ascertain the validity of solutions and steer clear of extraneous outcomes.