Problem 16 - Algebra 1

Question

Tell which equation you would use to isolate a variable. Explain your reasoning. $$\begin{aligned} &5 c+3 d=11\\\ &5 c-d=5 \end{aligned}$$

Step-by-Step Solution

Answer

The equation \(5c-d=5\) would be used to isolate the variable 'd'. The reasoning is that the coefficient of 'd' is 1 in this equation, which makes it easier to isolate. When we isolate 'd', we get \(d = 5c - 5\).

Step 1: Choice of Equation

Choosing the second equation \(5c - d = 5\) for isolating the variable 'd' as it has a coefficient of 1, which would make the process easier and straightforward.

Step 2: Isolating the Variable

To isolate d, move \(5c\) to the right-hand side by subtracting \(5c\) from both sides. The new equation becomes: \(-d=5-5c\).

Step 3: Expressing 'd' in Positive Form

To express 'd' in positive form multiply the equation by -1 to get rid of the negative sign in front of 'd'. This will give us: \(d = -5 + 5c\), which is same as: \(d = 5c - 5\).

Key Concepts

Linear Equations

Linear equations are equations where each term is either a constant or the product of a constant and a single variable. These types of equations can be identified by their highest variable power, which is always 1. Linear equations take the general form: \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable.
Linear equations are foundational in algebra because they represent a straight line when plotted on a graph. This makes them straightforward and easy to manipulate compared to non-linear equations, which include variables with powers greater than one.
Understanding linear equations is essential because they occur frequently in real-life situations. They form the basis for many mathematical problems and solutions in areas ranging from economics to engineering.

Variables Isolation

The process of isolating a variable within an equation involves arranging the equation so that the variable of interest appears by itself on one side of the equation. This process can be used to find the value of the variable if the rest of the terms are known. For instance, in the equation \( 5c - d = 5 \), the goal is to solve for \( d \) by isolating it.
To isolate \( d \), you need to move all the other terms to the opposite side of the equation. You can achieve this usually through methods such as:

• Adding or subtracting terms on both sides of the equation.
• Multiplying or dividing terms on both sides of the equation.

In the given example, subtracting \( 5c \) from both sides allowed for the isolation of \( d \). This is an essential step in solving linear equations and provides clarity by simplifying complex problems into more understandable parts.

Algebraic Manipulation

Algebraic manipulation refers to the various methods employed to rearrange and simplify equations. It involves performing appropriate operations to maintain the equality while making the equation simpler or more useful.
In the given problem, algebraic manipulation includes adjusting the equation \( 5c - d = 5 \) to solve for \( d \). By doing this, the equation was manipulated by subtracting \( 5c \) from both sides and then multiplying the entire equation by -1 to make \( d \) positive. These steps showcase how different algebraic operations can maintain the balance of the equations while effectively solving for the variables.
Effective algebraic manipulation often involves:

• Using operations such as addition, subtraction, multiplication, and division efficiently.
• Maintaining the equality of both sides of the equation throughout the manipulation process.
• Converting complex equations into simpler forms for better understanding and solution.

These techniques are central to solving mathematical problems efficiently and accurately.