Problem 34 - Introductory Algebra for College Students

Question

Solve each equation in using both the addition and multiplication properties of equality. Check proposed solutions. $$-3 y+4=13$$

Step-by-Step Solution

Answer

The solution to the equation \(-3 y + 4 = 13\) is \(y = -3\).

Step 1: Use the Addition Property of Equality

First, subtract 4 from both sides of the equation to isolate the term -3y. The equation becomes: \(-3 y = 13 - 4\). Simplifying the right-hand side will give us \(-3 y = 9\).

Step 2: Use the Multiplication Property of Equality

Next, divide both sides of the equation by -3 to isolate y. The equation thus becomes: \(y = 9 / -3\). Simplifying the right-hand side yields \(y = -3\).

Step 3: Check the Proposed Solution

Substitute y = -3 into the original equation. The original equation is \(-3 y + 4 = 13\). Substituting y = -3 gives \(-3*-3 + 4 = 13\). Simplifying the left-hand side gives \(9 + 4 = 13\), which simplified further gives \(13 = 13\). Since both sides of the equation are equal, our solution is correct.

Key Concepts

Addition Property of Equality

The Addition Property of Equality is a fundamental rule in solving equations. It states that you can add or subtract the same number from both sides of an equation, and the equation will remain balanced. Imagine a scale; if you add or take away the same weight from both sides, it stays equal or balanced. For our exercise, we applied this property first to isolate the term containing the variable.

• Original Equation: \(-3y + 4 = 13\)
• Action: Subtract 4 from both sides\(-3y + 4 - 4 = 13 - 4\)
• Simplified Result: \(-3y = 9\)

This step helped us move one step closer to isolating the variable \(y\), making it easier to solve.

Multiplication Property of Equality

The Multiplication Property of Equality allows us to multiply or divide both sides of an equation by the same non-zero number without changing the equality. This property is particularly useful for eliminating coefficients attached to variables. In our example, once we had the term \(-3y = 9\), we needed to isolate \(y\).

• Initial Step: Divide both sides by \(-3\).
• Equation Before Division: \(-3y = 9\)
• Perform Operation: \(y = \cfrac{9}{-3}\)
• Solution: \(y = -3\)

By applying the multiplication property, we successfully isolated \(y\) and found the solution to the equation.

Linear Equations

Linear equations, like \(-3y + 4 = 13\), involve variables raised to the power of one. These equations form straight lines when graphed and are often simple to solve using basic algebraic operations. In our case, we utilized both addition and multiplication properties to find the solution.

• Definition: An equation of the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is a variable.
• Our Example: \(-3y + 4 = 13\).
• Steps Used: Addition and multiplication properties helped us to isolate and solve for \(y\).
• Solution Check: Substituting \(y = -3\) back verified the correctness as both sides equaled \(13\).

Understanding linear equations is crucial as they form the foundation for more complex mathematical concepts.