Let \(a\) equal \(2 b+3 c-5 .\) What happens to the value of \(a\) if the value of \(b\) decreases by 1 and the value of \(c\) increases by 2\(?\) F. It increases by \(4 .\) G. It increases by 2 H. It increases by 1 . J. It is unchanged. K. It decreases by 2.

Algebra is like a game of rearranging and solving puzzles with numbers and letters. Algebraic manipulation is a fundamental skill that helps you change the form of an equation or expression to solve problems. It involves various techniques such as distributing, combining like terms, and factoring.

Let’s look at our example. The original exercise presents an equation and asks about the effect of changing the variables within it. The equation is given as \( a = 2b + 3c - 5 \). The question is what happens to \(a\) if \(b\) decreases by 1 and \(c\) increases by 2.

Applying algebraic manipulation, we adjust the equation to reflect these changes, which requires us to substitute the new values of \(b\) and \(c\) into the equation. When you’re dealing with a decrease or an increase, you're essentially adding or subtracting from the existing variable. This is a simple yet crucial form of algebraic manipulation. For instance,

• Increasing \(c\) by 2 becomes \(c+2\)
• Decreasing \(b\) by 1 becomes \(b-1\)

These new values, \(c' = c + 2\) and \(b' = b - 1\), are then used to find the new expression for \(a\) which is now \(a'\).

Cracking the code of an equation is at the heart of mathematics. Equation solving is the process of finding the values of variables that satisfy the equation. It's like finding the missing piece of a puzzle that makes the whole picture come together.

In our ACT Math problem, equation solving comes into play after we have completed the algebraic manipulation. We have the new equation: \( a' = 2(b - 1) + 3(c + 2) - 5 \). Solving this involves expanding and simplifying to see the new relationship between \(a'\) and \(a\).

When we simplify the equation—distributing the 2 to the terms inside the first parenthesis and 3 to the terms inside the second, followed by combining like terms—we unveil the new value of \(a'\) in terms of \(b\) and \(c\). This sort of manipulation and simplification is a central part of solving equations, leading us to the understanding that \(a'\) is actually 4 units greater than \(a\). The solution, therefore, inherently relies on equation solving to interpret the new variable relationships after the changes.

Variables are the alphabets of the mathematical language, and understanding their relationships is like understanding conversations in a story. Variable relationships explore how one variable affects another within equations and expressions. Identifying and working with these relationships is key to solving algebraic problems.

In our example, we examine how changes in the variables \(b\) and \(c\) affect the value of \(a\). By substituting \(b'\) and \(c'\) into the original equation and then simplifying it, we can evaluate the shift in value.

• The old equation is \( a = 2b + 3c - 5 \)
• The new equation becomes \( a' = 2b + 3c - 1 \)

By comparison, it's clear that \(a\) and \(a'\) differ by a constant number, here being 4, which signifies a direct relationship between the changes in \(b\) and \(c\) and the resultant change in \(a\). Through such analysis of variable relationships, we determine the precise impact of altering one or more variables on another, leading us to the correct answer in our ACT Math problem.