Placing Equations in the Table

Oct 13, 2025 by miodeya.com 23 views

Introduction

In mathematics, solving equations is a fundamental concept that helps us find the value of a variable. When we are given an equation with a variable, we need to isolate the variable to find its value. In this article, we will explore how to solve equations for a given variable and then place them in a table under the correct solution.

Understanding the Problem

The problem requires us to solve the equations for the given variable and then place them in a table under the correct solution. We are given two categories: $x=3$ and $x \neq 3$ . We need to solve the equations and place them in the correct category.

Solving Equations

To solve an equation, we need to isolate the variable. We can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value. Let's consider some examples:

Example 1: Solving for x

Suppose we have the equation $2x + 5 = 11$ . To solve for x, we need to isolate x. We can do this by subtracting 5 from both sides of the equation:

2x + 5 - 5 = 11 - 5

This simplifies to:

2x = 6

Now, we can divide both sides of the equation by 2 to solve for x:

\frac{2x}{2} = \frac{6}{2}

This simplifies to:

x = 3

So, the solution to the equation $2x + 5 = 11$ is $x = 3$ .

Example 2: Solving for x

Suppose we have the equation $x - 2 = 7$ . To solve for x, we need to isolate x. We can do this by adding 2 to both sides of the equation:

x - 2 + 2 = 7 + 2

This simplifies to:

x = 9

So, the solution to the equation $x - 2 = 7$ is $x = 9$ .

Example 3: Solving for x

Suppose we have the equation $x + 3 = 10$ . To solve for x, we need to isolate x. We can do this by subtracting 3 from both sides of the equation:

x + 3 - 3 = 10 - 3

This simplifies to:

x = 7

So, the solution to the equation $x + 3 = 10$ is $x = 7$ .

Placing Equations in the Table

Now that we have solved the equations, we can place them in the table under the correct solution. Let's consider the equations we solved earlier:

Equation	Solution
$2x + 5 = 11$	$x = 3$
$x - 2 = 7$	$x = 9$
$x + 3 = 10$	$x = 7$

We can see that the equations $2x + 5 = 11$ and $x + 3 = 10$ have solutions $x = 3$ and $x = 7$ , respectively. These equations belong to the category $x = 3$ .

On the other hand, the equation $x - 2 = 7$ has a solution $x = 9$ , which belongs to the category $x \neq 3$ .

Conclusion

In this article, we learned how to solve equations for a given variable and then place them in a table under the correct solution. We considered three examples of equations and solved them to find their solutions. We then placed the equations in the table under the correct solution. We hope this article has helped you understand how to solve equations and place them in a table under the correct solution.

Discussion Category: Mathematics

This article is part of the discussion category: mathematics. Mathematics is a branch of science that deals with numbers, quantities, and shapes. It is a fundamental subject that is used in various fields, including science, engineering, economics, and finance. Mathematics is a powerful tool that helps us understand the world around us and make informed decisions.

Key Takeaways

Solving equations is a fundamental concept in mathematics that helps us find the value of a variable.
To solve an equation, we need to isolate the variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
We can place equations in a table under the correct solution by solving the equations and then categorizing them based on their solutions.
Mathematics is a branch of science that deals with numbers, quantities, and shapes.
Mathematics is a fundamental subject that is used in various fields, including science, engineering, economics, and finance.

References

[1] Khan Academy. (n.d.). Solving Equations. Retrieved from https://www.khanacademy.org/math/algebra/solving-linear-equations
[2] Mathway. (n.d.). Solving Equations. Retrieved from https://www.mathway.com/subjects/solving-equations

Solving Equations for the Given Variable
Understanding the Problem
Solving Equations
Placing Equations in the Table
Conclusion
Discussion Category: Mathematics
Key Takeaways
References
Table of Contents
Frequently Asked Questions (FAQs) =====================================

Q: What is the purpose of solving equations?

A: The purpose of solving equations is to find the value of a variable. Equations are used to represent relationships between variables, and solving them helps us understand the relationships between the variables.

Q: How do I solve an equation?

A: To solve an equation, you need to isolate the variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same value. You can use inverse operations to isolate the variable.

Q: What is an inverse operation?

A: An inverse operation is an operation that undoes another operation. For example, addition and subtraction are inverse operations, as are multiplication and division.

Q: How do I know which operation to use to solve an equation?

A: To determine which operation to use, you need to look at the equation and identify the variable and the constant. You can then use inverse operations to isolate 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 is 1. A quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I solve a linear equation?

A: To solve a linear equation, you can use the following steps:

Simplify the equation by combining like terms.
Isolate the variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
Check your solution by plugging it back into the original equation.

Q: How do I solve a quadratic equation?

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

Simplify the equation by combining like terms.
Factor the equation, if possible.
Use the quadratic formula to solve the equation: x = (-b ± √(b^2 - 4ac)) / 2a
Check your solution by plugging it back into the original equation.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that is used to solve quadratic equations. It is: x = (-b ± √(b^2 - 4ac)) / 2a

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula. You can then simplify the expression and solve for x.

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. A single equation is a single equation that is solved independently.

Q: How do I solve a system of equations?

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

Simplify the equations by combining like terms.
Use substitution or elimination to solve the system of equations.
Check your solution by plugging it back into the original equations.

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

A: A linear inequality is an inequality in which the highest power of the variable is 1. A quadratic inequality is an inequality in which the highest power of the variable is 2.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you can use the following steps:

Simplify the inequality by combining like terms.
Isolate the variable by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.
Check your solution by plugging it back into the original inequality.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you can use the following steps:

Simplify the inequality by combining like terms.
Factor the inequality, if possible.
Use the quadratic formula to solve the inequality: x = (-b ± √(b^2 - 4ac)) / 2a
Check your solution by plugging it back into the original inequality.

Q: What is the difference between a rational expression and a polynomial expression?

A: A rational expression is an expression that is the ratio of two polynomials. A polynomial expression is an expression that is a sum of terms, each of which is a product of a variable and a constant.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you can use the following steps:

Factor the numerator and denominator, if possible.
Cancel out any common factors between the numerator and denominator.
Simplify the resulting expression.

Q: How do I add or subtract rational expressions?

A: To add or subtract rational expressions, you can use the following steps:

Factor the numerator and denominator, if possible.
Find a common denominator for the expressions.
Add or subtract the numerators and keep the common denominator.
Simplify the resulting expression.

Q: How do I multiply or divide rational expressions?

A: To multiply or divide rational expressions, you can use the following steps:

Factor the numerator and denominator, if possible.
Multiply or divide the numerators and denominators separately.
Simplify the resulting expression.

Q: What is the difference between a function and a relation?

A: A function is a relation in which each input corresponds to exactly one output. A relation is a set of ordered pairs that satisfy a certain condition.

Q: How do I determine if a relation is a function?

A: To determine if a relation is a function, you can use the following steps:

Check if each input corresponds to exactly one output.
Check if the relation is a one-to-one correspondence.

Q: What is the difference between a domain and a range?

A: The domain of a function is the set of all possible input values. The range of a function is the set of all possible output values.

Q: How do I find the domain and range of a function?

A: To find the domain and range of a function, you can use the following steps:

Check if the function is defined for all real numbers.
Check if the function is defined for a specific interval or set of values.
Check if the function has any restrictions or limitations.

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

A: A linear function is a function in which the highest power of the variable is 1. A quadratic function is a function in which the highest power of the variable is 2.

Q: How do I graph a linear function?

A: To graph a linear function, you can use the following steps:

Find the x-intercept and y-intercept of the function.
Plot the x-intercept and y-intercept on a coordinate plane.
Draw a line through the intercepts to graph the function.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you can use the following steps:

Find the x-intercepts and y-intercepts of the function.
Plot the x-intercepts and y-intercepts on a coordinate plane.
Draw a parabola through the intercepts to graph the function.

Q: What is the difference between a function and a graph?

A: A function is a mathematical object that assigns a value to each input. A graph is a visual representation of a function.

Q: How do I determine if a graph represents a function?

A: To determine if a graph represents a function, you can use the following steps:

Check if each input corresponds to exactly one output.
Check if the graph is a one-to-one correspondence.

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

A: A linear inequality is an inequality in which the highest power of the variable is 1. A quadratic inequality is an inequality in which the highest power of the variable is 2.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you can use the following steps:

Simplify the inequality by combining like terms.
Isolate the variable by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.
Check your solution by plugging it back into the original inequality.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you can use the following steps:

Simplify the inequality by combining like terms.
Factor the inequality, if possible.
Use the quadratic formula to solve the inequality: x = (-b ± √(b^2 - 4ac)) / 2a
Check your solution by plugging it back into the original inequality.

Q: What is the difference between a rational expression and a polynomial expression?

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you can use the following steps:

Factor the numerator and denominator, if possible.
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