Problem 36 - Algebra A Combined Function

Question

Solve each equation. See Examples 6 and \(7 .\) $$ -5(4 y-3)+2=-20 y+17 $$

Step-by-Step Solution

Answer

Infinite solutions; any value for \(y\) satisfies the equation.

Step 1: Distribute the Negative Five

Distribute the \(-5\) through the parentheses, multiplying it by each term inside. So, we have \(-5 \times 4y = -20y\) and \(-5 \times (-3) = +15\). The equation becomes:\[-20y + 15 + 2 = -20y + 17\]

Step 2: Simplify Both Sides

Combine like terms on the left side: \(15 + 2 = 17\). Now the equation is:\[-20y + 17 = -20y + 17\]

Step 3: Compare the Equations

Notice that the expressions on both sides of the equation \(-20y + 17\) are identical. This implies that the equation is true for all values of \(y\).

Step 4: Conclusion: Identify the Solution Type

Since both sides of the equation are identical, there is no specific value for \(y\) that needs to be solved; instead, any value of \(y\) will satisfy the equation.

Key Concepts

Distributive Property

The distributive property is a fundamental rule in algebra that allows you to simplify expressions when dealing with parentheses. In our example, we used the distributive property to multiply

• -5 by both terms inside the parentheses, which were -4y and -3.
• This gives us two individual products: -20y from (-5 imes 4y) and +15 from (-5 imes -3).

This process helps in expanding expressions, making it easier to simplify and solve equations. Learning to apply the distributive property correctly makes working with complex expressions much smoother.

Like Terms

Combining like terms is another key technique applied in the example. Like terms are terms in an algebraic expression that have identical variable parts or no variables (constants). In the sample equation,

• we grouped the constants 15 and 2 on the left side of the equation to simplify it. Combining them resulted in 17 as part of the equation.

By grouping and simplifying like terms, you reduce the complexity of an expression, helping you to better see the relationships and simplify the whole equation.

Infinite Solutions

In the example provided, we arrived at a scenario where both sides of the equation were the same: -20y + 17 = -20y + 17. This tells us something special.
When both sides of the equation are identical, it means that the equation is always true, regardless of what value we assign to the variable (y in this case).
This indicates what we call ‘infinite solutions’.

• This means there isn't just one number or a few numbers that make the equation work.
• Instead, any possible value for (y) will satisfy the equation.

It’s an interesting situation because it showcases how equations can sometimes have more open-ended solution sets.

Equation Solving Process

The process of solving equations unfolds systematically. It typically involves several steps which often include:

• Applying the distributive property to remove parentheses, if any.
• Combining like terms to simplify the expression on both sides.
• Analyzing the resulting simplified equation.
• Drawing conclusions about the potential solutions, whether it’s a specific number, a range, or infinite solutions.

In our example, after distributing and simplifying, we discovered the identical sides scenario. The careful execution of each step helps in understanding and resolving the equation efficiently. By practicing these steps, you become adept at tackling a wide variety of algebraic challenges.