Problem 51 - Algebra A Combined Function

Question

To avoid fractions, which of the equations below would you use if solving for \(y ?\) Explain why. a. \(\frac{1}{2} x-4 y=\frac{3}{4}\) b. \(8 x-5 y=13\) c. \(7 x-y=19\)

Step-by-Step Solution

Answer

Use equation c: \(7x - y = 19\) to avoid fractions.

Step 1: Identify Fraction in Each Equation

First, examine equation a: \( \frac{1}{2}x - 4y = \frac{3}{4} \). It has fractions \( \frac{1}{2} \) and \( \frac{3}{4} \). Moving on to equation b: \( 8x - 5y = 13 \), it has no fractions. Finally, equation c: \( 7x - y = 19 \), also has no fractions.

Step 2: Choose Equations Without Fractions

To avoid fractions when solving for \( y \), we should select equations that do not have any fractional coefficients or constants. Equations b and c both meet this criterion as they involve only whole numbers.

Step 3: Determine Simplest Equation for Solving

Between equation b: \( 8x - 5y = 13 \) and equation c: \( 7x - y = 19 \), equation c is simpler because the coefficient of \( y \) is 1. Solving for \( y \) will be straightforward since dividing or multiplying by 1 is not necessary.

Step 4: Select Equation c for Solution

Since equation c is simpler due to \( y \) having a coefficient of 1, it would be easier to isolate \( y \) on one side without introducing additional arithmetic steps, thus avoiding fractions efficiently.

Key Concepts

Avoiding Fractions

When solving linear equations, working without fractions often makes the process easier. Fractions can complicate calculations and introduce additional steps. In the equations given as examples, equation a includes fractions, while b and c do not. By choosing equations b or c, you simplify your task. Without fractions:

• Calculations are straightforward, reducing chances of errors.
• You don’t need to find a common denominator, which can be time-consuming.
• Arithmetic steps involve simpler numbers, making the solution process quicker.

Given our options, working with whole numbers in equations like b and c is more efficient.

Linear Equation Coefficients

The coefficients in a linear equation are the numbers that multiply the variables. They're crucial because they determine how we manipulate the equation to solve for a variable. In equation c, for instance, the coefficient of y is 1. This makes solving for y straightforward since dividing or multiplying by 1 does not change the value. Meanwhile, equation b's coefficient for y is -5, which requires an extra step of division when isolating y.

• Equations with a coefficient of 1 for the variable of interest (e.g., y) are generally preferred.
• They minimize the arithmetic involved in solving the equation.
• Simplicity in coefficients aids in a clearer understanding of the manipulation needed.

Always consider the coefficients when deciding which equation form is the easiest to solve.

Isolating Variables

Isolating a variable means getting the variable by itself on one side of the equation. This is usually the goal when solving equations. With equation c, isolating y is particularly easy, thanks to its coefficient being 1. Here's the simple process:

• We take the original equation, 7x - y = 19.
• To isolate y, move 7x to the other side: -y = -7x + 19.
• Change the signs to solve for y: y = 7x - 19.

Isolating variables efficiently minimizes errors and simplifies the further steps in solving equations.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to find the value of a variable. This process includes adding, subtracting, multiplying, or dividing terms. With equation b, algebraic manipulation involves more steps compared to equation c:

• Equation b: 8x - 5y = 13 requires us to isolate y by subtracting 8x and then dividing each term by -5.
• In contrast, equation c, as shown earlier, involves a simple rearrangement to directly solve for y.
• This involves fewer arithmetic operations, reducing complexity and risk of mistakes.

When manipulating algebraic equations, always aim to use methods that simplify your work and decrease unnecessary calculations.