Solve The Following Equations:1. $\frac{1}{4}x + \frac{5}{3}x - 4 = 2 - \frac{1}{12}x$2. $3.2b - 4.7 = 3b - 3.3$3. $4.6y - Y + 4 = Y - 1.2$4. 7.5 C − 2.5 C + 8 = − 7 7.5c - 2.5c + 8 = -7 7.5 C − 2.5 C + 8 = − 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 linear equations, each with a unique set of coefficients and constants. We will use algebraic techniques to isolate the variable and find the solution.

Equation 1: $\frac{1}{4}x + \frac{5}{3}x - 4 = 2 - \frac{1}{12}x$

To solve this equation, we need to combine like terms and isolate the variable x. The first step is to eliminate the fractions by finding the least common multiple (LCM) of the denominators.

The LCM of 4, 3, and 12 is 12.

We can rewrite the equation as:

$$\frac{3}{12}x + \frac{20}{12}x - 4 = 2 - \frac{1}{12}x$$

Combine like terms:

$$\frac{23}{12}x - 4 = 2 - \frac{1}{12}x$$

Add $\frac{1}{12}x$ to both sides:

$$\frac{23}{12}x + \frac{1}{12}x - 4 = 2$$

Combine like terms:

$$\frac{24}{12}x - 4 = 2$$

Simplify:

$$2x - 4 = 2$$

Add 4 to both sides:

$$2x = 6$$

Divide both sides by 2:

$$x = 3$$

Equation 2: $3.2b - 4.7 = 3b - 3.3$

To solve this equation, we need to isolate the variable b. The first step is to add 4.7 to both sides:

$$3.2b = 3b + 3.3 - 4.7$$

Combine like terms:

$$3.2b = 3b - 1.4$$

Subtract 3b from both sides:

$$0.2b = -1.4$$

Divide both sides by 0.2:

$$b = -7$$

Equation 3: $4.6y - y + 4 = y - 1.2$

To solve this equation, we need to combine like terms and isolate the variable y. The first step is to add y to both sides:

$$4.6y + 4 = 2y - 1.2$$

Subtract 2y from both sides:

$$2.6y + 4 = -1.2$$

Subtract 4 from both sides:

$$2.6y = -5.2$$

Divide both sides by 2.6:

$$y = -2$$

Equation 4: $7.5c - 2.5c + 8 = -7$

To solve this equation, we need to combine like terms and isolate the variable c. The first step is to add 2.5c to both sides:

$$10c + 8 = -7$$

Subtract 8 from both sides:

$$10c = -15$$

Divide both sides by 10:

$$c = -1.5$$

Conclusion

Solving linear equations requires a step-by-step approach, and each equation has its unique set of coefficients and constants. By combining like terms, isolating the variable, and using algebraic techniques, we can find the solution to each equation. In this article, we solved four linear equations, each with a different set of coefficients and constants. We hope that this article has provided a clear and concise guide to solving linear equations.

Tips and Tricks

• Always start by combining like terms and isolating the variable.
• Use algebraic techniques such as addition, subtraction, multiplication, and division to solve the equation.
• Check your solution by plugging it back into the original equation.
• Practice, practice, practice! Solving linear equations is a skill that requires practice to develop.

Real-World Applications

Linear equations have numerous real-world applications, including:

• Physics: Linear equations are used to describe the motion of objects, including velocity, acceleration, and distance.
• Engineering: Linear equations are used to design and optimize systems, including electrical circuits, mechanical systems, and structural systems.
• Economics: Linear equations are used to model economic systems, including supply and demand, cost-benefit analysis, and optimization.
• Computer Science: Linear equations are used in computer graphics, game development, and machine learning.

Final Thoughts

Solving linear equations is a fundamental skill that is essential for success in mathematics, science, and engineering. By following the step-by-step approach outlined in this article, you can develop the skills and confidence to solve linear equations with ease. Remember to practice regularly, and don't be afraid to ask for help when you need it. With persistence and dedication, you can master the art of solving linear equations and unlock a world of possibilities.

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 performing algebraic operations such as addition, subtraction, multiplication, and division. You can also use techniques such as combining like terms and using inverse operations.

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, 2x + 3 = 5 is a linear equation, while x^2 + 4x + 4 = 0 is a quadratic equation.

Q: Can I use a calculator to solve a linear equation?

A: Yes, you can use a calculator to solve a linear equation. However, it's always a good idea to check your solution by plugging it back into the original equation to ensure that it's correct.

Q: How do I know if a linear equation has a solution?

A: A linear equation has a solution if it is consistent, meaning that the equation is true for at least one value of the variable. If the equation is inconsistent, it means that there is no value of the variable that can make the equation true.

Q: Can I solve a linear equation with multiple variables?

A: Yes, you can solve a linear equation with multiple variables. However, it's often more complicated than solving a linear equation with a single variable. You may need to use techniques such as substitution or elimination to solve the equation.

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 or more variables, while a system of linear equations is a set of two or more linear equations with the same variables. Solving a system of linear equations requires finding the values of the variables that satisfy all of the equations in the system.

Q: Can I use technology to solve a system of linear equations?

A: Yes, you can use technology such as graphing calculators or computer software to solve a system of linear equations. However, it's always a good idea to check your solution by plugging it back into the original equations to ensure that it's correct.

Q: How do I know if a system of linear equations has a solution?

A: A system of linear equations has a solution if it is consistent, meaning that the equations are true for at least one value of the variables. If the system is inconsistent, it means that there is no value of the variables that can make the equations true.

Q: Can I solve a system of linear equations using substitution or elimination?

A: Yes, you can solve a system of linear equations using substitution or elimination. Substitution involves solving one equation for one variable and then substituting that expression into the other equation. Elimination involves adding or subtracting the equations to eliminate one of the variables.

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

A: A linear equation has a highest power of 1, while a nonlinear equation has a highest power greater than 1. For example, 2x + 3 = 5 is a linear equation, while x^2 + 4x + 4 = 0 is a nonlinear equation.

Q: Can I use a calculator to solve a nonlinear equation?

A: Yes, you can use a calculator to solve a nonlinear equation. However, it's often more complicated than solving a linear equation, and you may need to use numerical methods or approximation techniques to find the solution.

Q: How do I know if a nonlinear equation has a solution?

A: A nonlinear equation may have one or more solutions, depending on the specific equation and the values of the variables. You can use numerical methods or approximation techniques to find the solution, or you can use algebraic techniques such as factoring or completing the square to solve the equation.

Conclusion

Solving linear equations and systems of linear equations is a fundamental skill that is essential for success in mathematics, science, and engineering. By following the step-by-step approach outlined in this article, you can develop the skills and confidence to solve linear equations with ease. Remember to practice regularly, and don't be afraid to ask for help when you need it. With persistence and dedication, you can master the art of solving linear equations and unlock a world of possibilities.