Solve The Equation. Justify Each Step And Check Your Solution.3) $\frac{m}{5} = -4$4) $3p = -\frac{2}{3}$5) 2.5 R = 75 2.5r = 75 2.5 R = 75

Introduction

Solving equations is a fundamental concept in mathematics that involves isolating the variable on one side of the equation. In this article, we will focus on solving three different equations: $\frac{m}{5} = -4$, $3p = -\frac{2}{3}$, and $2.5r = 75$. We will justify each step and check our solution to ensure that we have found the correct value of the variable.

Solving the Equation $\frac{m}{5} = -4$

To solve the equation $\frac{m}{5} = -4$, we need to isolate the variable $m$. We can do this by multiplying both sides of the equation by 5.

Step 1: Multiply Both Sides by 5

Multiplying both sides of the equation by 5 will eliminate the fraction and allow us to solve for $m$.

$$\frac{m}{5} = -4$$

$$m = -4 \times 5$$

$$m = -20$$

Step 2: Check the Solution

To check our solution, we can substitute $m = -20$ back into the original equation and verify that it is true.

$$\frac{m}{5} = -4$$

$$\frac{-20}{5} = -4$$

$$-4 = -4$$

Since the equation is true, we can be confident that our solution is correct.

Solving the Equation $3p = -\frac{2}{3}$

To solve the equation $3p = -\frac{2}{3}$, we need to isolate the variable $p$. We can do this by dividing both sides of the equation by 3.

Step 1: Divide Both Sides by 3

Dividing both sides of the equation by 3 will eliminate the coefficient and allow us to solve for $p$.

$$3p = -\frac{2}{3}$$

$$p = -\frac{2}{3} \div 3$$

$$p = -\frac{2}{3} \times \frac{1}{3}$$

$$p = -\frac{2}{9}$$

Step 2: Check the Solution

To check our solution, we can substitute $p = -\frac{2}{9}$ back into the original equation and verify that it is true.

$$3p = -\frac{2}{3}$$

$$3 \times -\frac{2}{9} = -\frac{2}{3}$$

$$-\frac{6}{9} = -\frac{2}{3}$$

$$-\frac{2}{3} = -\frac{2}{3}$$

Since the equation is true, we can be confident that our solution is correct.

Solving the Equation $2.5r = 75$

To solve the equation $2.5r = 75$, we need to isolate the variable $r$. We can do this by dividing both sides of the equation by 2.5.

Step 1: Divide Both Sides by 2.5

Dividing both sides of the equation by 2.5 will eliminate the coefficient and allow us to solve for $r$.

$$2.5r = 75$$

$$r = 75 \div 2.5$$

$$r = 30$$

Step 2: Check the Solution

To check our solution, we can substitute $r = 30$ back into the original equation and verify that it is true.

$$2.5r = 75$$

$$2.5 \times 30 = 75$$

$$75 = 75$$

Since the equation is true, we can be confident that our solution is correct.

Conclusion

In this article, we solved three different equations: $\frac{m}{5} = -4$, $3p = -\frac{2}{3}$, and $2.5r = 75$. We justified each step and checked our solution to ensure that we had found the correct value of the variable. By following the steps outlined in this article, you can solve similar equations and become more confident in your ability to solve mathematical problems.

Frequently Asked Questions

• Q: What is the first step in solving an equation? A: The first step in solving an equation is to isolate the variable on one side of the equation.
• Q: How do I check my solution? A: To check your solution, substitute the value of the variable back into the original equation and verify that it is true.
• Q: What if I get a different solution? A: If you get a different solution, recheck your work and make sure that you have followed the steps correctly.

Final Thoughts

Solving equations is a fundamental concept in mathematics that involves isolating the variable on one side of the equation. By following the steps outlined in this article, you can solve similar equations and become more confident in your ability to solve mathematical problems. Remember to always check your solution to ensure that you have found the correct value of the variable.

Introduction

Solving equations is a fundamental concept in mathematics that involves isolating the variable on one side of the equation. In this article, we will answer some frequently asked questions about solving equations, including how to check your solution, what to do if you get a different solution, and more.

Q: What is the first step in solving an equation?

A: The first step in solving an equation is to isolate the variable on one side of the equation. This can be done by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Q: How do I check my solution?

A: To check your solution, substitute the value of the variable back into the original equation and verify that it is true. This will ensure that you have found the correct value of the variable.

Q: What if I get a different solution?

A: If you get a different solution, recheck your work and make sure that you have followed the steps correctly. It's also possible that the original equation is incorrect or that there is a mistake in the problem.

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 2x + 3 = 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 = (-b ± √(b^2 - 4ac)) / 2a. This formula will give you two possible solutions for the variable.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when solving an equation. The order of operations is:

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

Q: How do I simplify an expression?

A: To simplify an expression, combine like terms and eliminate any unnecessary operations. For example, the expression 2x + 3 + 2x can be simplified to 4x + 3.

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

A: A variable is a value that can change, while a constant is a value that remains the same. For example, in the equation 2x + 3 = 5, x is a variable because its value can change, while 3 is a constant because its value remains the same.

Q: How do I graph an equation?

A: To graph an equation, you can use a graphing calculator or graph paper. Plot points on the graph and connect them with a line to create a graph of the equation.

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

A: A function is a relation in which each input value corresponds to exactly one output value, while a relation is a set of ordered pairs that may or may not be a function. For example, the equation y = 2x is a function because each input value corresponds to exactly one output value, while the equation y = x^2 is a relation because each input value corresponds to two output values.

Conclusion

Solving equations is a fundamental concept in mathematics that involves isolating the variable on one side of the equation. By following the steps outlined in this article, you can solve similar equations and become more confident in your ability to solve mathematical problems. Remember to always check your solution to ensure that you have found the correct value of the variable.

Final Thoughts

Solving equations is a skill that takes practice to develop. With patience and persistence, you can become proficient in solving equations and apply this skill to a wide range of mathematical problems.

Additional Resources

• For more information on solving equations, check out the following resources:

• Khan Academy: Solving Equations
• Mathway: Solving Equations
• Wolfram Alpha: Solving Equations

Related Articles

• Solving Linear Equations
• Solving Quadratic Equations
• Graphing Equations
• Functions and Relations