Question

If \(15-3 b=21\), what is the value of \(5-b\) ?

Short answer

The value of \(5-b\) is 7.

Step-by-step solution

Solve for b in the given equation

We are given the equation: \(15-3 b=21\). To solve for \(b\), first subtract 15 from both sides of the equation: \[-3b = 21 - 15\] \[-3b = 6\] Now, divide by -3 on both sides to find the value of \(b\): \[b = \frac{6}{-3}\] \[b = -2\]

Calculate the value of 5 - b

Now that we have the value of \(b\) (-2), we can calculate the value of \(5-b\): \[5-b = 5 - (-2)\] Remember that subtracting a negative number is the same as adding the positive version of that number: \[5-b = 5 + 2\] \[5-b = 7\] Therefore, the value of \(5-b\) is 7.

Key concepts

Linear Equations

A linear equation is a type of algebraic equation where each term is either a constant or the product of a constant and a single variable. Linear equations are usually in the form of \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants. In our original exercise, the equation \( 15 - 3b = 21 \) is already structured in a linear form.
Solving linear equations is an essential skill in algebra, as it helps you find the value of unknown variables. These equations often represent real-world problems where relationships between quantities are linear or have a constant rate of change.
By following step-by-step procedures, such as isolating the variable on one side of the equation, we can systematically find the solution. Understanding this foundational concept enables students to solve more complex equations in future maths explorations.

Problem Solving

Problem solving in algebra involves identifying what the problem is asking for and determining the best strategy to find the solution. This often involves breaking the problem down into smaller, more manageable steps.
In our exercise, the problem effectively asked us to first find the value of \( b \) and then use that information to compute \( 5-b \). By tackling the problem step by step, students can avoid feeling overwhelmed and can focus on solving one part of the problem at a time.
Common strategies in problem solving include identifying what you already know, determining what you need to find out, and looking for patterns or relationships in the information given. Recognizing these elements will guide you toward your solution, making problem solving a systematic and approachable process.

Basic Arithmetic

Basic arithmetic forms the backbone of solving algebraic problems. It involves operations like addition, subtraction, multiplication, and division, which were essential in our exercise.
First, we used subtraction to simplify the equation \( 15 - 3b = 21 \) by subtracting 15 from both sides. This gave us \(-3b = 6\). Then, we divided both sides by \(-3\) to solve for \( b \). Later, recognizing that subtracting a negative is equivalent to addition helped us find \( 5-b = 7 \).
As simple as these operations may seem, practicing basic arithmetic skills is crucial to ensuring precision and accuracy. Mistakes often arise from miscalculations, so being comfortable with these operations is key to successful algebra problem solving.

Equation Solving

Equation solving is a cornerstone of algebra, where the goal is to find the value of an unknown that satisfies a given equation. The steps to solve include: simplifying the equation, isolating the variable, and performing basic arithmetic operations.
In the provided equation \( 15 - 3b = 21 \), we simplified by moving terms and dividing coefficients to isolate and solve for \( b \). Finding \( b = -2 \) allowed us to substitute back into the expression \( 5-b \) to find its value.
Solving equations effectively requires a clear strategy and careful handling of algebraic rules. This helps ensure that solutions are correct and equations balance properly, reflecting a deep understanding of the mathematical relationships involved.