How to Determine whether a Given Number is a Solution to an Equation

Oct 13, 2025 by thesweetdesigns.com 50 views

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Introduction

In mathematics, an equation is a statement that two mathematical expressions are equal. A solution to an equation is a value that makes the equation true. In this article, we will discuss how to determine whether a given number is a solution to a given equation. We will use the equation $$x + 12 = 19$$ as an example to illustrate the concept.

What is a Solution to an Equation?

A solution to an equation is a value that makes the equation true. In other words, if we substitute the solution into the equation, the equation will be satisfied. For example, if we have the equation $$x + 5 = 10$$, then the solution is $$x = 5$$, because when we substitute $$x = 5$$ into the equation, we get $$5 + 5 = 10$$, which is true.

How to Determine whether a Given Number is a Solution to an Equation

To determine whether a given number is a solution to an equation, we need to substitute the number into the equation and check if the equation is satisfied. If the equation is satisfied, then the number is a solution to the equation. If the equation is not satisfied, then the number is not a solution to the equation.

Example: Is -7 a Solution to the Equation $$x + 12 = 19$$?

Let's use the equation $$x + 12 = 19$$ as an example to illustrate how to determine whether a given number is a solution to an equation. We are given the number -7 and we need to determine whether it is a solution to the equation.

To determine whether -7 is a solution to the equation, we need to substitute -7 into the equation and check if the equation is satisfied. We get:

-7 + 12 = 19

To evaluate the left-hand side of the equation, we need to add -7 and 12. We get:

-7 + 12 = 5

Since $$5 \neq 19$$, the equation is not satisfied. Therefore, -7 is not a solution to the equation $$x + 12 = 19$$.

Conclusion

In conclusion, to determine whether a given number is a solution to a given equation, we need to substitute the number into the equation and check if the equation is satisfied. If the equation is satisfied, then the number is a solution to the equation. If the equation is not satisfied, then the number is not a solution to the equation. We used the equation $$x + 12 = 19$$ as an example to illustrate the concept and determined that -7 is not a solution to the equation.

Frequently Asked Questions

Q: What is a solution to an equation?

A: A solution to an equation is a value that makes the equation true.

Q: How do I determine whether a given number is a solution to an equation?

A: To determine whether a given number is a solution to an equation, you need to substitute the number into the equation and check if the equation is satisfied.

Q: What if the equation is not satisfied?

A: If the equation is not satisfied, then the number is not a solution to the equation.

Step-by-Step Solution

Step 1: Substitute the given number into the equation

Substitute -7 into the equation $$x + 12 = 19$$.

Step 2: Evaluate the left-hand side of the equation

Evaluate the left-hand side of the equation $$-7 + 12$$.

Step 3: Check if the equation is satisfied

Check if $$5 = 19$$.

Step 4: Determine whether the given number is a solution to the equation

Since $$5 \neq 19$$, the equation is not satisfied. Therefore, -7 is not a solution to the equation $$x + 12 = 19$$.

Example Problems

Problem 1: Is 3 a solution to the equation $$x - 2 = 5$$?

Substitute 3 into the equation $$x - 2 = 5$$.

Evaluate the left-hand side of the equation $$3 - 2$$.

Check if $$1 = 5$$.

Since $$1 \neq 5$$, the equation is not satisfied. Therefore, 3 is not a solution to the equation $$x - 2 = 5$$.

Problem 2: Is -2 a solution to the equation $$x + 3 = 1$$?

Substitute -2 into the equation $$x + 3 = 1$$.

Evaluate the left-hand side of the equation $$-2 + 3$$.

Check if $$1 = 1$$.

Since $$1 = 1$$, the equation is satisfied. Therefore, -2 is a solution to the equation $$x + 3 = 1$$.

Final Answer

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Frequently Asked Questions

Q: What is a solution to an equation?

A: A solution to an equation is a value that makes the equation true.

Q: How do I determine whether a given number is a solution to an equation?

A: To determine whether a given number is a solution to an equation, you need to substitute the number into the equation and check if the equation is satisfied.

Q: What if the equation is not satisfied?

A: If the equation is not satisfied, then the number is not a solution to the equation.

Q: Can a number be a solution to more than one equation?

A: Yes, a number can be a solution to more than one equation. For example, the number 5 is a solution to the equation $$x + 2 = 7$$ and also a solution to the equation $$x - 3 = 2$$.

Q: Can an equation have more than one solution?

A: Yes, an equation can have more than one solution. For example, the equation $$x^2 = 4$$ has two solutions: $$x = 2$$ and $$x = -2$$.

Q: How do I find the solutions to an equation?

A: To find the solutions to an equation, you need to isolate the variable on one side of the equation. You can use algebraic operations such as addition, subtraction, multiplication, and division to isolate the variable.

Q: What if the equation has a variable in the denominator?

A: If the equation has a variable in the denominator, you need to get rid of the variable in the denominator before you can isolate the variable. You can do this by multiplying both sides of the equation by the denominator.

Q: Can I use a calculator to find the solutions to an equation?

A: Yes, you can use a calculator to find the solutions to an equation. However, you need to make sure that the calculator is set to the correct mode and that you are using the correct operations.

Step-by-Step Solutions

Step 1: Substitute the given number into the equation

Substitute 0 into the equation $$x + 5 = 5$$.

Step 2: Evaluate the left-hand side of the equation

Evaluate the left-hand side of the equation $$0 + 5$$.

Step 3: Check if the equation is satisfied

Check if $$5 = 5$$.

Step 4: Determine whether the given number is a solution to the equation

Since $$5 = 5$$, the equation is satisfied. Therefore, 0 is a solution to the equation $$x + 5 = 5$$.

Step 1: Substitute the given number into the equation

Substitute -3 into the equation $$x - 2 = 1$$.

Step 2: Evaluate the left-hand side of the equation

Evaluate the left-hand side of the equation $$-3 - 2$$.

Step 3: Check if the equation is satisfied

Check if $$-5 = 1$$.

Step 4: Determine whether the given number is a solution to the equation

Since $$-5 \neq 1$$, the equation is not satisfied. Therefore, -3 is not a solution to the equation $$x - 2 = 1$$.

Example Problems

Problem 1: Is 4 a solution to the equation $$x + 2 = 6$$?

Substitute 4 into the equation $$x + 2 = 6$$.

Evaluate the left-hand side of the equation $$4 + 2$$.

Check if $$6 = 6$$.

Since $$6 = 6$$, the equation is satisfied. Therefore, 4 is a solution to the equation $$x + 2 = 6$$.

Problem 2: Is -4 a solution to the equation $$x - 2 = 2$$?

Substitute -4 into the equation $$x - 2 = 2$$.

Evaluate the left-hand side of the equation $$-4 - 2$$.

Check if $$-6 = 2$$.

Since $$-6 \neq 2$$, the equation is not satisfied. Therefore, -4 is not a solution to the equation $$x - 2 = 2$$.

Final Answer

In conclusion, to determine whether a given number is a solution to a given equation, we need to substitute the number into the equation and check if the equation is satisfied. If the equation is satisfied, then the number is a solution to the equation. If the equation is not satisfied, then the number is not a solution to the equation. We used various examples to illustrate the concept and determined whether different numbers are solutions to different equations.