Solve The Following Equations:1. \[$6b - 4 = 38\$\]2. \[$8m + 11 = -53\$\]3. \[$-\frac{1}{4}c - 5 = 20\$\]4. \[$1.2y + 0.52 = 7\$\]

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving four different linear equations, each with a unique variable and coefficient. We will use a step-by-step approach to solve each equation, and provide explanations and examples to help illustrate the concepts.

Equation 1: Solving for b

The Equation

$$6b - 4 = 38$$

Step 1: Add 4 to Both Sides

To isolate the variable b, we need to get rid of the constant term -4 on the left-hand side of the equation. We can do this by adding 4 to both sides of the equation.

$$6b - 4 + 4 = 38 + 4$$

Step 2: Simplify the Equation

After adding 4 to both sides, the equation becomes:

$$6b = 42$$

Step 3: Divide Both Sides by 6

To solve for b, we need to get rid of the coefficient 6 on the left-hand side of the equation. We can do this by dividing both sides of the equation by 6.

$$\frac{6b}{6} = \frac{42}{6}$$

Step 4: Simplify the Equation

After dividing both sides by 6, the equation becomes:

$$b = 7$$

Equation 2: Solving for m

The Equation

$$8m + 11 = -53$$

Step 1: Subtract 11 from Both Sides

To isolate the variable m, we need to get rid of the constant term 11 on the left-hand side of the equation. We can do this by subtracting 11 from both sides of the equation.

$$8m + 11 - 11 = -53 - 11$$

Step 2: Simplify the Equation

After subtracting 11 from both sides, the equation becomes:

$$8m = -64$$

Step 3: Divide Both Sides by 8

To solve for m, we need to get rid of the coefficient 8 on the left-hand side of the equation. We can do this by dividing both sides of the equation by 8.

$$\frac{8m}{8} = \frac{-64}{8}$$

Step 4: Simplify the Equation

After dividing both sides by 8, the equation becomes:

$$m = -8$$

Equation 3: Solving for c

The Equation

$$-\frac{1}{4}c - 5 = 20$$

Step 1: Add 5 to Both Sides

To isolate the variable c, we need to get rid of the constant term -5 on the left-hand side of the equation. We can do this by adding 5 to both sides of the equation.

$$-\frac{1}{4}c - 5 + 5 = 20 + 5$$

Step 2: Simplify the Equation

After adding 5 to both sides, the equation becomes:

$$-\frac{1}{4}c = 25$$

Step 3: Multiply Both Sides by -4

To solve for c, we need to get rid of the coefficient -1/4 on the left-hand side of the equation. We can do this by multiplying both sides of the equation by -4.

$$\left(-\frac{1}{4}c\right) \times (-4) = 25 \times (-4)$$

Step 4: Simplify the Equation

After multiplying both sides by -4, the equation becomes:

$$c = -100$$

Equation 4: Solving for y

The Equation

$$1.2y + 0.52 = 7$$

Step 1: Subtract 0.52 from Both Sides

To isolate the variable y, we need to get rid of the constant term 0.52 on the left-hand side of the equation. We can do this by subtracting 0.52 from both sides of the equation.

$$1.2y + 0.52 - 0.52 = 7 - 0.52$$

Step 2: Simplify the Equation

After subtracting 0.52 from both sides, the equation becomes:

$$1.2y = 6.48$$

Step 3: Divide Both Sides by 1.2

To solve for y, we need to get rid of the coefficient 1.2 on the left-hand side of the equation. We can do this by dividing both sides of the equation by 1.2.

$$\frac{1.2y}{1.2} = \frac{6.48}{1.2}$$

Step 4: Simplify the Equation

After dividing both sides by 1.2, the equation becomes:

$$y = 5.4$$

Conclusion

Solving linear equations is a crucial skill for students and professionals alike. In this article, we have solved four different linear equations, each with a unique variable and coefficient. We have used a step-by-step approach to solve each equation, and provided explanations and examples to help illustrate the concepts. By following these steps, you can solve linear equations with ease and confidence.

Additional Resources

• Khan Academy: Linear Equations
• Mathway: Linear Equations Solver
• Wolfram Alpha: Linear Equations Calculator

Frequently Asked Questions

• Q: What is a linear equation? A: A linear equation is an equation in which the highest power of the variable(s) is 1.
• Q: How do I solve a linear equation? A: To solve a linear equation, you need to isolate the variable by getting rid of the constant term and the coefficient.
• Q: What is the difference between a linear equation and a quadratic equation? A: A linear equation has a highest power of 1, while a quadratic equation has a highest power of 2.
  Frequently Asked Questions: Linear Equations =====================================================

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 by getting rid of the constant term and the coefficient. Here are the steps:

1. Add or subtract the same value to both sides of the equation to get rid of the constant term.
2. Multiply or divide both sides of the equation by the same value to get rid of the coefficient.
3. Simplify the equation to get the final solution.

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

A: A linear equation has a highest power of 1, while a quadratic equation has a highest power of 2. For example:

Linear equation: 2x + 3 = 5 Quadratic equation: x^2 + 4x + 4 = 0

Q: How do I graph a linear equation?

A: To graph a linear equation, you need to find two points on the line and plot them on a coordinate plane. You can use the slope-intercept form of the equation (y = mx + b) to find the slope and y-intercept.

Q: What is the slope-intercept form of a linear equation?

A: The slope-intercept form of a linear equation is:

y = mx + b

where m is the slope and b is the y-intercept.

Q: How do I find the slope of a linear equation?

A: To find the slope of a linear equation, you need to use the formula:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are two points on the line.

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

A: A linear equation is a single equation with one variable, while a system of linear equations is a set of two or more equations with two or more variables.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you need to use one of the following methods:

1. Substitution method: Substitute one equation into the other equation to solve for one variable.
2. Elimination method: Add or subtract the two equations to eliminate one variable.
3. Graphing method: Graph the two equations on a coordinate plane and find the point of intersection.

Q: What is the importance of linear equations in real-life applications?

A: Linear equations are used in a wide range of real-life applications, including:

1. Physics: Linear equations are used to describe the motion of objects and the forces acting on them.
2. Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
3. Economics: Linear equations are used to model economic systems and make predictions about future trends.
4. Computer Science: Linear equations are used in algorithms and data structures to solve problems and make decisions.

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

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

1. Not following the order of operations (PEMDAS).
2. Not isolating the variable correctly.
3. Not checking the solution for extraneous solutions.
4. Not using the correct method for solving the equation.

Q: How can I practice solving linear equations?

A: You can practice solving linear equations by:

1. Using online resources, such as Khan Academy and Mathway.
2. Working on practice problems and exercises.
3. Using a calculator or computer program to check your solutions.
4. Joining a study group or working with a tutor to get help and feedback.