Solve The System Of Equations:(1) { \frac{x-1}{4} = \frac{y}{6}$}$(2) { \frac{z}{3} = Y = 9$}$

Introduction

In mathematics, a system of equations is a set of equations that are related to each other through a common variable or variables. Solving a system of equations involves finding the values of the variables that satisfy all the equations in the system. In this article, we will focus on solving a system of two equations with three variables: x, y, and z.

The System of Equations

The system of equations we will be solving is:

1. $\frac{x-1}{4} = \frac{y}{6}$
2. $\frac{z}{3} = y = 9$

Step 1: Simplify the First Equation

To simplify the first equation, we can start by cross-multiplying both sides of the equation:

$$6(x-1) = 4y$$

Expanding the left-hand side of the equation, we get:

$$6x - 6 = 4y$$

Rearranging the equation to isolate y, we get:

$$4y = 6x - 6$$

Dividing both sides of the equation by 4, we get:

$$y = \frac{6x - 6}{4}$$

Step 2: Simplify the Second Equation

The second equation is already simplified, so we can move on to the next step.

Step 3: Substitute the Expression for y into the Second Equation

Now that we have an expression for y in terms of x, we can substitute this expression into the second equation:

$$\frac{z}{3} = \frac{6x - 6}{4} = 9$$

Cross-multiplying both sides of the equation, we get:

$$4z = 3(6x - 6)$$

Expanding the right-hand side of the equation, we get:

$$4z = 18x - 18$$

Dividing both sides of the equation by 4, we get:

$$z = \frac{18x - 18}{4}$$

Step 4: Solve for x

Now that we have expressions for y and z in terms of x, we can substitute these expressions into one of the original equations to solve for x. Let's use the first equation:

$$\frac{x-1}{4} = \frac{y}{6}$$

Substituting the expression for y in terms of x, we get:

$$\frac{x-1}{4} = \frac{6x - 6}{6}$$

Cross-multiplying both sides of the equation, we get:

$$6(x-1) = 4(6x - 6)$$

Expanding the left-hand side of the equation, we get:

$$6x - 6 = 24x - 24$$

Rearranging the equation to isolate x, we get:

$$18x = 18$$

Dividing both sides of the equation by 18, we get:

$$x = 1$$

Step 5: Find the Values of y and z

Now that we have the value of x, we can find the values of y and z by substituting x into the expressions we derived earlier:

$$y = \frac{6x - 6}{4} = \frac{6(1) - 6}{4} = 0$$

$$z = \frac{18x - 18}{4} = \frac{18(1) - 18}{4} = 0$$

Conclusion

In this article, we solved a system of two equations with three variables: x, y, and z. We started by simplifying the first equation and then substituted the expression for y into the second equation. We then solved for x and used the value of x to find the values of y and z. The final answer is x = 1, y = 0, and z = 0.

Final Answer

Introduction

In our previous article, we solved a system of two equations with three variables: x, y, and z. In this article, we will provide a Q&A guide to help you understand the concepts and steps involved in solving a system of equations.

Q: What is a system of equations?

A: A system of equations is a set of equations that are related to each other through a common variable or variables. Solving a system of equations involves finding the values of the variables that satisfy all the equations in the system.

Q: How do I know if I have a system of equations?

A: You have a system of equations if you have two or more equations that involve the same variables. For example:

1. $\frac{x-1}{4} = \frac{y}{6}$
2. $\frac{z}{3} = y = 9$

In this example, we have two equations that involve the variables x, y, and z.

Q: What are the steps involved in solving a system of equations?

A: The steps involved in solving a system of equations are:

1. Simplify the first equation
2. Substitute the expression for y into the second equation
3. Solve for x
4. Find the values of y and z

Q: What is cross-multiplication?

A: Cross-multiplication is a technique used to eliminate variables in a system of equations. It involves multiplying both sides of an equation by a common factor to eliminate one of the variables.

Q: How do I use cross-multiplication to solve a system of equations?

A: To use cross-multiplication to solve a system of equations, follow these steps:

1. Multiply both sides of the first equation by the denominator of the second equation
2. Multiply both sides of the second equation by the denominator of the first equation
3. Simplify the resulting equations
4. Solve for the remaining variable

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

A: A linear equation is an equation in which the highest power of the variable is 1. For example:

$$x + 2y = 3$$

A nonlinear equation is an equation in which the highest power of the variable is greater than 1. For example:

$$x^2 + 2y = 3$$

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

A: Yes, you can use substitution to solve a system of equations. Substitution involves substituting the expression for one variable into the other equation to eliminate that variable.

Q: What are some common mistakes to avoid when solving a system of equations?

A: Some common mistakes to avoid when solving a system of equations include:

• Not simplifying the equations before solving
• Not using cross-multiplication or substitution correctly
• Not checking the solutions to make sure they satisfy all the equations

Conclusion

In this article, we provided a Q&A guide to help you understand the concepts and steps involved in solving a system of equations. We covered topics such as what a system of equations is, how to know if you have a system of equations, and how to use cross-multiplication and substitution to solve a system of equations. We also discussed common mistakes to avoid when solving a system of equations.

Final Tips

• Make sure to simplify the equations before solving
• Use cross-multiplication or substitution correctly
• Check the solutions to make sure they satisfy all the equations

By following these tips and practicing solving systems of equations, you will become more confident and proficient in solving these types of problems.