Multiplying both sides by the wrong denominator

Oct 13, 2025 by windailys.com 63 views

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Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific type of linear equation, namely the equation $$\frac{x-3}{4}=8$$. We will break down the solution process into manageable steps and provide a clear explanation of each step.

Understanding the Equation

The given equation is $$\frac{x-3}{4}=8$$. To solve for $$x$$, we need to isolate the variable $$x$$ on one side of the equation. The equation involves a fraction, which can be eliminated by multiplying both sides of the equation by the denominator, in this case, $$4$$.

Step 1: Multiply Both Sides by the Denominator

To eliminate the fraction, we multiply both sides of the equation by the denominator, which is $$4$$. This gives us:

\frac{x-3}{4} \times 4 = 8 \times 4

Simplifying the equation, we get:

x - 3 = 32

Step 2: Add 3 to Both Sides

To isolate the variable $$x$$, we need to get rid of the constant term $$-3$$ on the left side of the equation. We can do this by adding $$3$$ to both sides of the equation:

x - 3 + 3 = 32 + 3

Simplifying the equation, we get:

x = 35

Conclusion

Final Answer

The final answer to the equation $$\frac{x-3}{4}=8$$ is:

This is the correct answer among the given options A, B, C, and D.

Why is this the Correct Answer?

The correct answer is 35 because it satisfies the equation $$\frac{x-3}{4}=8$$. When we substitute $$x = 35$$ into the equation, we get:

\frac{35-3}{4} = \frac{32}{4} = 8

This shows that the equation is true when $$x = 35$$, making it the correct answer.

What if I Made a Mistake?

If you made a mistake while solving the equation, don't worry! You can always go back and recheck your work. Here are some common mistakes to watch out for:

Multiplying both sides by the wrong denominator: Make sure to multiply both sides by the correct denominator, which is $$4$$ in this case.
Adding or subtracting the wrong number: Double-check that you are adding or subtracting the correct number to both sides of the equation.
Not simplifying the equation: Make sure to simplify the equation after each step to avoid confusion.

Practice Makes Perfect

Solving linear equations takes practice, so don't be discouraged if you make mistakes at first. With time and practice, you will become more confident and proficient in solving linear equations.

Common Linear Equations

Here are some common linear equations that you may encounter:

$\frac{x+2}{3}=5$
$\frac{x-1}{2}=3$
$\frac{x+4}{5}=2$

Try solving these equations on your own, and then check your answers with the solutions provided below.

Solutions to Common Linear Equations

Here are the solutions to the common linear equations listed above:

$\frac{x+2}{3}=5$$: $$x = 13$
$\frac{x-1}{2}=3$$: $$x = 7$
$\frac{x+4}{5}=2$$: $$x = -2$

Conclusion

In this article, we solved the linear equation $$\frac{x-3}{4}=8$$ by following a step-by-step process. We eliminated the fraction by multiplying both sides of the equation by the denominator, and then isolated the variable $$x$$ by adding $$3$$ to both sides of the equation. The final solution is $$x = 35$$. We also provided solutions to common linear equations and offered tips for practicing and avoiding common mistakes. With time and practice, you will become more confident and proficient in solving linear equations.

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Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. It can be written in the form $$ax + b = c$$, where $$a$$, $$b$$, and $$c$$ are constants, and $$x$$ is the variable.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable on one side of the equation. You can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Q: What is the order of operations when solving a linear equation?

A: When solving a linear equation, you should follow the order of operations (PEMDAS):

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 eliminate fractions when solving a linear equation?

A: To eliminate fractions when solving a linear equation, you can multiply both sides of the equation by the denominator. This will eliminate the fraction and allow you to solve for the variable.

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(s) is 1, while a quadratic equation is an equation in which the highest power of the variable(s) is 2. For example, $$x + 2 = 3$$ is a linear equation, while $$x^2 + 4x + 4 = 0$$ is a quadratic equation.

Q: How do I determine if an equation is linear or quadratic?

A: To determine if an equation is linear or quadratic, you can look at the highest power of the variable(s). If the highest power is 1, the equation is linear. If the highest power is 2, the equation is quadratic.

Q: What are some common mistakes to avoid when solving linear equations?

A: Some common mistakes to avoid when solving linear equations include:

Multiplying both sides by the wrong denominator: Make sure to multiply both sides by the correct denominator.
Adding or subtracting the wrong number: Double-check that you are adding or subtracting the correct number to both sides of the equation.
Not simplifying the equation: Make sure to simplify the equation after each step to avoid confusion.

Q: How can I practice solving linear equations?

A: You can practice solving linear equations by working through example problems, such as those found in a math textbook or online resource. You can also try solving linear equations on your own, and then check your answers with the solutions provided.

Q: What are some real-world applications of linear equations?

A: Linear equations have many real-world applications, including:

Finance: Linear equations can be used to calculate interest rates, investment returns, and other financial metrics.
Science: Linear equations can be used to model the behavior of physical systems, such as the motion of objects or the flow of fluids.
Engineering: Linear equations can be used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Q: How can I use technology to solve linear equations?

A: You can use technology, such as calculators or computer software, to solve linear equations. Many calculators and computer programs have built-in functions for solving linear equations, making it easy to find the solution.

Q: What are some common types of linear equations?

A: Some common types of linear equations include:

Slope-intercept form: $$y = mx + b$$, where $$m$$ is the slope and $$b$$ is the y-intercept.
Standard form: $$ax + by = c$$, where $$a$$, $$b$$, and $$c$$ are constants.
Point-slope form: $$y - y_1 = m(x - x_1)$$, where $$m$$ is the slope and $$(x_1, y_1)$$ is a point on the line.

Q: How can I graph a linear equation?

A: You can graph a linear equation by using a coordinate plane and plotting points that satisfy the equation. You can also use technology, such as a graphing calculator or computer software, to graph a linear equation.

Q: What are some common applications of linear equations in real life?

A: Linear equations have many real-world applications, including:

Designing and optimizing systems: Linear equations can be used to design and optimize systems, such as bridges, buildings, and electronic circuits.
Modeling the behavior of physical systems: Linear equations can be used to model the behavior of physical systems, such as the motion of objects or the flow of fluids.
Calculating interest rates and investment returns: Linear equations can be used to calculate interest rates and investment returns.

Q: How can I use linear equations to solve problems in real life?

A: You can use linear equations to solve problems in real life by applying the concepts and techniques learned in this article. Some examples of how to use linear equations to solve problems in real life include:

Designing and optimizing systems: Use linear equations to design and optimize systems, such as bridges, buildings, and electronic circuits.
Modeling the behavior of physical systems: Use linear equations to model the behavior of physical systems, such as the motion of objects or the flow of fluids.
Calculating interest rates and investment returns: Use linear equations to calculate interest rates and investment returns.

Q: What are some common challenges when solving linear equations?

A: Some common challenges when solving linear equations include:

Multiplying both sides by the wrong denominator: Make sure to multiply both sides by the correct denominator.
Adding or subtracting the wrong number: Double-check that you are adding or subtracting the correct number to both sides of the equation.
Not simplifying the equation: Make sure to simplify the equation after each step to avoid confusion.

Q: How can I overcome common challenges when solving linear equations?

A: You can overcome common challenges when solving linear equations by:

Double-checking your work: Make sure to double-check your work to avoid mistakes.
Using technology: Use technology, such as calculators or computer software, to solve linear equations.
Practicing regularly: Practice solving linear equations regularly to build your skills and confidence.

Q: What are some common mistakes to avoid when graphing linear equations?

A: Some common mistakes to avoid when graphing linear equations include:

Not using a coordinate plane: Make sure to use a coordinate plane when graphing a linear equation.
Not plotting enough points: Make sure to plot enough points to get a clear picture of the graph.
Not using technology: Use technology, such as a graphing calculator or computer software, to graph a linear equation.

Q: How can I use linear equations to solve problems in science and engineering?

A: You can use linear equations to solve problems in science and engineering by applying the concepts and techniques learned in this article. Some examples of how to use linear equations to solve problems in science and engineering include:

Modeling the behavior of physical systems: Use linear equations to model the behavior of physical systems, such as the motion of objects or the flow of fluids.
Designing and optimizing systems: Use linear equations to design and optimize systems, such as bridges, buildings, and electronic circuits.
Calculating interest rates and investment returns: Use linear equations to calculate interest rates and investment returns.

Q: What are some common applications of linear equations in finance?

A: Linear equations have many applications in finance, including:

Calculating interest rates and investment returns: Linear equations can be used to calculate interest rates and investment returns.
Modeling the behavior of financial systems: Linear equations can be used to model the behavior of financial systems, such as the stock market or the economy.
Designing and optimizing financial systems: Linear equations can be used to design and optimize financial systems, such as investment portfolios or retirement plans.

Q: How can I use linear equations to solve problems in finance?

A: You can use linear equations to solve problems in finance by applying the concepts and techniques learned in this article. Some examples of how to use linear equations to solve problems in finance include:

Calculating interest rates and investment returns: Use linear equations to calculate interest rates and investment returns.
Modeling the behavior of financial systems: Use linear equations to model the behavior of financial systems, such as the stock market or the economy.
Designing and optimizing financial systems: Use linear equations to design and optimize financial systems, such as investment portfolios or retirement plans.