Problem 14 - Algebra 1: Concepts and Skills

Question

Tell which equation you would use to isolate a variable. Explain. $$ \begin{array}{r} {4 a+3 b=-5} \\ {a-b=-3} \end{array} $$

Step-by-Step Solution

Answer

The second equation, \(a - b = -3\), would be the better choice to isolate a variable. By rearranging, you get \(a = b - 3\).

Step 1: Identify the equations

Two equations are given: \(4a + 3b = -5\) and \(a - b = -3\). The task is to choose one equation that can help to isolate a variable.

Step 2: Choose the equation

The second equation, \(a - b = -3\), would be a better choice in this scenario to isolate a variable. The reasoning behind this is that in the second equation, the coefficients before \(a\) and \(b\) are 1 and -1 respectively, which makes it easier to isolate a variable. On contrary, the first equation has the coefficients 4 and 3 for \(a\) and \(b\) respectively, which would require more steps to isolate a variable.

Step 3: Isolate the variable

The goal is to isolate the variable \(a\). By rearranging the equation \(a - b = -3\), it's easy to see that \(a = b - 3\).

Key Concepts

Linear Equations

Linear equations are like straight roads in the world of mathematics.
They can be drawn as straight lines on a graph and usually take the form of \(ax + by = c\).
In our original exercise, we have two linear equations: \(4a + 3b = -5\) and \(a - b = -3\). These equations involve variables \(a\) and \(b\), with coefficients that are constants.Understanding linear equations is crucial because they commonly appear in different areas of mathematics and real-life problems.
They are called 'linear' because they form straight lines when graphed, reflecting a constant rate of change. Linear equations help us find relationships between variables, and in many scenarios, we need to solve them or rearrange them to find the value of one of the variables. This process can be interesting and fun when you understand the right steps to take.

Equation Rewriting

Equation rewriting is an important skill that allows you to rearrange an equation to better understand or solve it.
The key is to manipulate the equation without changing its truthfulness.When faced with equations, like our example of \(a - b = -3\), we might need to isolate one variable to solve the problem.
In this context, rewriting means rearranging the equation to have one variable alone on one side of the equation.

• For example, from the equation \(a - b = -3\), we can rearrange it to \(a = b - 3\).
• This step makes it easier to understand the relationship between \(a\) and \(b\).

Choosing the simplest path is important when rewriting; simpler equations are easier to work with, saving time and reducing errors.

Variable Manipulation

Variable manipulation involves turning the equation around, almost like solving a puzzle.
It's about isolating one variable to see clearly what it equals. In the given exercise, isolating \(a\) from the equation \(a - b = -3\) requires adding or subtracting from both sides of the equation.
By adding \(b\) to both sides, we isolate \(a\):\[a - b + b = -3 + b\a = b - 3\]This process involves simple arithmetic operations:

• Adding or subtracting terms to both sides
• Multiplying or dividing both sides by a non-zero number

These actions maintain the equality of the equation so you can solve it step-by-step.
When manipulating variables, always perform the same operation on both sides of the equation. This ensures that the equation remains balanced, like a perfectly leveled scale.