Problem 32 - Elementary Algebra

Question

For the following problems, solve the linear equations in two variables. $$ b=4 a-12, \text { if } a=-7 $$

Step-by-Step Solution

Answer

Question: Given the linear equation \(b = 4a - 12\) and the value of 'a' as -7, find the value of 'b'. Answer: The value of 'b' when 'a' is equal to -7 in the given equation is -40.

Step 1: Write down the given equation and the value of 'a'

We are given the equation \(b = 4a - 12\), and the value of 'a' is \(a = -7\).

Step 2: Substitute the value of 'a' into the equation

Substitute \(a = -7\) into the equation \(b= 4a - 12\). So we have: \(b = 4(-7) - 12\).

Step 3: Multiply

Now, we multiply 4 and -7: \(b = -28 -12\).

Step 4: Subtract

Finally, subtract 12 from -28: \(b = -40\).

Step 5: Write down the solution

The solution for the given linear equation with the value of 'a' as -7 is \(b = -40\).

Key Concepts

Solving Equations

Solving equations is like uncovering a mystery hidden in numbers. It involves finding the value of one or more unknowns that make the equation true. In our problem, we're tasked with finding out what the variable 'b' is when 'a' has a specific value. This process follows a systematic approach, ensuring each step logically follows the last.

When solving equations, you'll often start with an expression that includes the unknowns. By performing operations such as substitution, addition, subtraction, multiplication, or division, the equation can be simplified to uncover the values of these unknowns.

The key idea is maintaining balance. Whatever operation you do on one side of the equation, you must do it to the other side to keep it equal. In the given exercise, once we know 'a', we're able to substitute and simplify, leading us to the solution where 'b' equals -40.

Two Variables

In this exercise, our equation involves two variables: 'a' and 'b'. A variable can be thought of as a placeholder for a number we do not yet know. Linear equations in two variables usually take the form of \(ax + by = c\), but they may be presented differently, as seen in our problem.

These types of equations represent straight lines when plotted on a graph. Here, the equation expresses a relationship between 'a' and 'b'. This means as 'a' changes, 'b' adjusts accordingly to maintain the equation's truth. This exercise helps us understand these relationships by solving for one variable's value when the other is known.

Understanding how these variables interact is crucial, not only in mathematics but also in describing real-world situations mathematically. Whether it's balancing chemical equations or calculating financial data, grasping how to handle two-variable equations is an essential skill.

Substitution Method

The substitution method is a powerful technique used for solving systems of equations, especially when one of the equations is already solved for a variable. In our problem, we're given that \(b = 4a - 12\), and we're instructed to use \(a = -7\).

Substitution involves taking the known value (in this case, \(a = -7\)) and substituting it into the equation to find the other unknown variable. This allows us to transform the equation into one with a single variable, making it easier to solve.

Here’s a simple process for using the substitution method:

• Identify the variable with a known value or solved equation.
• Substitute this value into the other equation(s).
• Solve the resulting equation to find the unknown variable.

This step-by-step technique helps to streamline the solving process, especially useful in more complex systems involving multiple equations.