Problem 68 - Algebra 1: Concepts and Skills

Question

Solve the equation. \(-2+4 c=19\)

Step-by-Step Solution

Answer

The solution to the equation is \(c = 21 / 4\), which further simplifies to \(c = 5.25\).

Step 1: Move constant to the other side

First, let's get rid of the constant on the left side. To do this, we add 2 to both sides of the equation. That gives: \(-2 + 2 + 4c = 19 + 2\), which simplifies to \(4c = 21\).

Step 2: Solve for c

Next, we need to isolate \( c \). To do this, we divide both sides of the equation by 4. That gives: \(c = 21 / 4\).

Key Concepts

Solving Equations

In mathematics, equations are like a bridge that helps us find the unknown values. Solving equations involves finding the solution that makes the equation true. For a linear equation such as \[-2 + 4c = 19\], our goal is to determine the value of \(c\) that satisfies the equation. Linear equations can often look complex initially, but with a structured approach, they become manageable.

• Always remember to perform the same operation on both sides of the equation. This keeps the equation balanced.
• Identify constants and coefficients. Constants are plain numbers, and coefficients are the numbers in front of variables.
• Work step by step. Do not rush through, as small mistakes can lead to incorrect results.

Solving equations is a fundamental skill in algebra and by practicing, you become more familiar with patterns and techniques.

Algebraic Manipulation

Algebraic manipulation refers to rearranging and simplifying equations using algebraic rules and operations. This is crucial when solving equations, as it helps us systematically tackle each part of the equation.
For the equation \[-2 + 4c = 19\], our first step in manipulation is to add 2 to both sides to eliminate the constant from the left. Why do we add exactly 2? Because \(-2 + 2\) results in 0, effectively removing the constant term on that side.

• Addition and subtraction are often used to move constant terms from one side of the equation to another.
• Multiplication and division are useful for handling coefficients of variables, like turning \(4c\) into \(c\).

Practice is key. The more you manipulate equations, the easier it becomes to recognize which operations to use.

Isolation of Variables

The process of isolation focuses on getting the desired variable on one side of the equation by itself. This is essential because it directly provides the solution.For instance, when we have \(4c = 21\), our goal is to isolate \(c\). To achieve this, we divide both sides by 4. This step is crucial because the 4 in \(4c\) is a coefficient that needs to be eliminated to leave \(c\) alone.

• Always perform the inverse operation. Here, the coefficient is multiplied, so we use division.
• Check your work. After isolating the variable, it's helpful to substitute back into the original equation to ensure it's correct.
• Remember that isolation of variables is not limited to division. Depending on the equation, you might add, subtract, or multiply too.

Being systematic in your approach to isolation makes it easier to avoid missteps and ensures accuracy in your solutions.