Making Variables the Subject of Formulas

Oct 13, 2025 by miodeya.com 126 views

Introduction

In physics and algebra, formulas are used to describe the relationships between variables. However, sometimes we need to isolate a specific variable to solve a problem or understand a concept. In this article, we will explore how to make variables the subject of formulas and solve for them in various equations.

Making Variables the Subject of Formulas

1. Making ${$ a$}$ the Subject of the Formula ${$ v = u + at$}$

To make ${$ a$}$ the subject of the formula ${$ v = u + at$}$, we need to isolate ${$ a$}$ on one side of the equation. We can do this by subtracting ${$ u$}$ from both sides and then dividing by ${$ t$}$.

Step 1: Subtract ${$ u$}$ from both sides

${$ v = u + at$}$

Subtract ${$ u$}$ from both sides:

${$ v - u = at$}$

Step 2: Divide both sides by ${$ t$}$

${$ v - u = at$}$

Divide both sides by ${$ t$}$:

${$ a = \frac{v - u}{t}$}$

Therefore, the formula ${$ v = u + at$}$ can be rewritten with ${$ a$}$ as the subject:

${$ a = \frac{v - u}{t}$}$

2. Making ${$ s$}$ the Subject of the Formula ${$ v^2 = u^2 + 2as$}$

To make ${$ s$}$ the subject of the formula ${$ v^2 = u^2 + 2as$}$, we need to isolate ${$ s$}$ on one side of the equation. We can do this by subtracting ${$ u^2$}$ from both sides and then dividing by ${ $2a\$}$ .

Step 1: Subtract ${$ u^2$}$ from both sides

${$ v^2 = u^2 + 2as$}$

Subtract ${$ u^2$}$ from both sides:

${$ v^2 - u^2 = 2as$}$

Step 2: Divide both sides by ${ $2a\$}$

${$ v^2 - u^2 = 2as$}$

Divide both sides by ${ $2a\$}$ :

${$ s = \frac{v^2 - u^2}{2a}$}$

Therefore, the formula ${$ v^2 = u^2 + 2as$}$ can be rewritten with ${$ s$}$ as the subject:

${$ s = \frac{v^2 - u^2}{2a}$}$

Solving for ${$ x$}$ in Algebraic Equations

1. Solving for ${$ x$}$ in the Equation ${$ ax + b = c$}$

To solve for ${$ x$}$ in the equation ${$ ax + b = c$}$, we need to isolate ${$ x$}$ on one side of the equation. We can do this by subtracting ${$ b$}$ from both sides and then dividing by ${$ a$}$.

Step 1: Subtract ${$ b$}$ from both sides

${$ ax + b = c$}$

Subtract ${$ b$}$ from both sides:

${$ ax = c - b$}$

Step 2: Divide both sides by ${$ a$}$

${$ ax = c - b$}$

Divide both sides by ${$ a$}$:

${$ x = \frac{c - b}{a}$}$

Therefore, the equation ${$ ax + b = c$}$ can be solved for ${$ x$}$:

${$ x = \frac{c - b}{a}$}$

2. Solving for ${$ x$}$ in the Equation ${$ x + a = b$}$

To solve for ${$ x$}$ in the equation ${$ x + a = b$}$, we need to isolate ${$ x$}$ on one side of the equation. We can do this by subtracting ${$ a$}$ from both sides.

Step 1: Subtract ${$ a$}$ from both sides

${$ x + a = b$}$

Subtract ${$ a$}$ from both sides:

${$ x = b - a$}$

Therefore, the equation ${$ x + a = b$}$ can be solved for ${$ x$}$:

${$ x = b - a$}$

Conclusion

In this article, we have explored how to make variables the subject of formulas and solve for them in various equations. We have seen how to isolate ${$ a$}$ in the formula ${$ v = u + at$}$ and ${$ s$}$ in the formula ${$ v^2 = u^2 + 2as$}$. We have also seen how to solve for ${$ x$}$ in the equations ${$ ax + b = c$}$ and ${$ x + a = b$}$. By following these steps, we can easily isolate variables and solve for them in a variety of equations.

References

[1] Khan Academy. (n.d.). Algebra. Retrieved from https://www.khanacademy.org/math/algebra
[2] Mathway. (n.d.). Algebra Solver. Retrieved from https://www.mathway.com/algebra

Note

Q: What is the difference between a variable and a constant in an equation?

A: In an equation, a variable is a value that can change, while a constant is a value that remains the same. For example, in the equation ${$ x + 2 = 5$}$, ${$ x$}$ is a variable because its value can change, while ${ $2\$}$ and ${ $5\$}$ are constants because their values remain the same.

Q: How do I know which variable to isolate in an equation?

A: To determine which variable to isolate, you need to identify the variable that is being asked for in the problem. For example, if the problem asks for the value of ${$ x$}$ in the equation ${ $2x + 3 = 7\$}$ , you would isolate ${$ x$}$ by subtracting ${ $3\$}$ from both sides and then dividing by ${ $2\$}$ .

Q: What is the order of operations in solving an equation?

A: The order of operations in solving an equation is:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I handle fractions in an equation?

A: To handle fractions in an equation, you need to follow the same steps as solving a regular equation, but with the added complexity of fractions. For example, if the equation is ${$ \frac{x}{2} + 3 = 5$}$, you would first subtract ${ $3\$}$ from both sides to get ${$ \frac{x}{2} = 2$}$, and then multiply both sides by ${ $2\$}$ to get ${$ x = 4$}$.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2. For example, the equation ${$ x + 2 = 5$}$ is a linear equation, while the equation ${$ x^2 + 4x + 4 = 0$}$ is a quadratic equation.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula:

${$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$}$

where ${$ a$}$, ${$ b$}$, and ${$ c$}$ are the coefficients of the quadratic equation.

Q: What is the difference between a system of equations and a single equation?

A: A system of equations is a set of two or more equations that are solved simultaneously, while a single equation is a single equation that is solved independently. For example, the system of equations ${$ x + 2y = 4$}$ and ${ $2x - 3y = -3\$}$ is a system of equations, while the single equation ${$ x + 2y = 4$}$ is a single equation.

Q: How do I solve a system of equations?

A: To solve a system of equations, you can use the following methods:

Substitution: Substitute the expression for one variable from one equation into the other equation.
Elimination: Add or subtract the equations to eliminate one variable.
Graphing: Graph the equations on a coordinate plane and find the point of intersection.

Conclusion

In this article, we have answered some of the most frequently asked questions about solving equations. We have covered topics such as variables and constants, order of operations, fractions, linear and quadratic equations, and systems of equations. By following these tips and techniques, you can become more confident and proficient in solving equations and solving problems in mathematics.

References

[1] Khan Academy. (n.d.). Algebra. Retrieved from https://www.khanacademy.org/math/algebra
[2] Mathway. (n.d.). Algebra Solver. Retrieved from https://www.mathway.com/algebra

Note

The questions and answers in this article are based on the principles of algebra and mathematics. The steps and techniques outlined in this article are general and can be applied to a variety of equations and problems. However, it is always a good idea to check your work and verify the solutions using a calculator or other tools.