Writing an Equivalent Equation without Denominators

Oct 13, 2025 by windailys.com 56 views

Introduction

In mathematics, a system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables. These equations can be represented in various forms, including fractions, decimals, and integers. In this article, we will focus on writing an equivalent equation without denominators for the first equation in a system of linear equations and then solving the system.

Writing an Equivalent Equation without Denominators

The first equation in the system of linear equations is $\frac{x}{2} + \frac{y}{3} = 6$ . To write an equivalent equation without denominators, we need to eliminate the fractions. We can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, which is 6.

$\frac{x}{2} + \frac{y}{3} = 6$

Multiply both sides by 6:

$3x + 2y = 36$

Now, we have an equivalent equation without denominators.

Solving the System of Linear Equations

We have two equations:

$3x + 2y = 36$ ... (1)

$x - y = 2$ ... (2)

We can solve this system of linear equations using the method of substitution or elimination. In this case, we will use the elimination method.

Step 1: Multiply Equation (2) by 2

To eliminate the variable y, we need to multiply equation (2) by 2.

$x - y = 2$

Multiply both sides by 2:

$2x - 2y = 4$

Step 2: Add Equation (1) and the Resulting Equation

Now, we can add equation (1) and the resulting equation to eliminate the variable y.

$3x + 2y = 36$

$2x - 2y = 4$

Add both equations:

$5x = 40$

Step 3: Solve for x

Now, we can solve for x by dividing both sides of the equation by 5.

$5x = 40$

Divide both sides by 5:

$x = 8$

Step 4: Substitute x into Equation (2)

Now that we have the value of x, we can substitute it into equation (2) to solve for y.

$x - y = 2$

Substitute x = 8:

$8 - y = 2$

Step 5: Solve for y

Now, we can solve for y by subtracting 8 from both sides of the equation.

$8 - y = 2$

Subtract 8 from both sides:

$-y = -6$

Multiply both sides by -1:

$y = 6$

Conclusion

In this article, we wrote an equivalent equation without denominators for the first equation in a system of linear equations and then solved the system using the elimination method. We found that the values of x and y are 8 and 6, respectively.

Introduction

Writing an Equivalent Equation without Denominators

$\frac{x}{2} + \frac{y}{3} = 6$

Multiply both sides by 6:

$3x + 2y = 36$

Now, we have an equivalent equation without denominators.

Solving the System of Linear Equations

We have two equations:

$3x + 2y = 36$ ... (1)

$x - y = 2$ ... (2)

We can solve this system of linear equations using the method of substitution or elimination. In this case, we will use the elimination method.

Step 1: Multiply Equation (2) by 2

To eliminate the variable y, we need to multiply equation (2) by 2.

$x - y = 2$

Multiply both sides by 2:

$2x - 2y = 4$

Step 2: Add Equation (1) and the Resulting Equation

Now, we can add equation (1) and the resulting equation to eliminate the variable y.

$3x + 2y = 36$

$2x - 2y = 4$

Add both equations:

$5x = 40$

Step 3: Solve for x

Now, we can solve for x by dividing both sides of the equation by 5.

$5x = 40$

Divide both sides by 5:

$x = 8$

Step 4: Substitute x into Equation (2)

Now that we have the value of x, we can substitute it into equation (2) to solve for y.

$x - y = 2$

Substitute x = 8:

$8 - y = 2$

Step 5: Solve for y

Now, we can solve for y by subtracting 8 from both sides of the equation.

$8 - y = 2$

Subtract 8 from both sides:

$-y = -6$

Multiply both sides by -1:

$y = 6$

Conclusion

Introduction

Writing an Equivalent Equation without Denominators

$\frac{x}{2} + \frac{y}{3} = 6$

Multiply both sides by 6:

$3x + 2y = 36$

Now, we have an equivalent equation without denominators.

Solving the System of Linear Equations

We have two equations:

$3x + 2y = 36$ ... (1)

$x - y = 2$ ... (2)

We can solve this system of linear equations using the method of substitution or elimination. In this case, we will use the elimination method.

Step 1: Multiply Equation (2) by 2

To eliminate the variable y, we need to multiply equation (2) by 2.

$x - y = 2$

Multiply both sides by 2:

$2x - 2y = 4$

Step 2: Add Equation (1) and the Resulting Equation

Now, we can add equation (1) and the resulting equation to eliminate the variable y.

$3x + 2y = 36$

$2x - 2y = 4$

Add both equations:

$5x = 40$

Step 3: Solve for x

Now, we can solve for x by dividing both sides of the equation by 5.

$5x = 40$

Divide both sides by 5:

$x = 8$

Step 4: Substitute x into Equation (2)

Now that we have the value of x, we can substitute it into equation (2) to solve for y.

$x - y = 2$

Substitute x = 8:

$8 - y = 2$

Step 5: Solve for y

Now, we can solve for y by subtracting 8 from both sides of the equation.

$8 - y = 2$

Subtract 8 from both sides:

$-y = -6$

Multiply both sides by -1:

$y = 6$

Conclusion

Introduction

Q&A: Frequently Asked Questions

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables.

Q: How do I write an equivalent equation without denominators?

A: To write an equivalent equation without denominators, you need to eliminate the fractions. You can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.

Q: What is the least common multiple (LCM) of the denominators?

A: The LCM of the denominators is the smallest number that is a multiple of all the denominators.

Q: How do I solve a system of linear equations using the elimination method?

A: To solve a system of linear equations using the elimination method, you need to eliminate one of the variables by adding or subtracting the equations.

Q: What is the elimination method?

A: The elimination method is a technique used to solve a system of linear equations by eliminating one of the variables.

Q: How do I multiply an equation by a constant?

A: To multiply an equation by a constant, you need to multiply both sides of the equation by the constant.

Q: What is the difference between the substitution method and the elimination method?

A: The substitution method involves substituting one equation into another to solve for one variable, while the elimination method involves eliminating one variable by adding or subtracting the equations.

Q: Can I use the elimination method to solve a system of linear equations with fractions?

A: Yes, you can use the elimination method to solve a system of linear equations with fractions by multiplying both sides of the equation by the LCM of the denominators.

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

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

Not multiplying both sides of the equation by the LCM of the denominators
Not eliminating one of the variables
Not checking the solution for consistency

Q: How do I check the solution for consistency?

A: To check the solution for consistency, you need to substitute the values of the variables into both equations and check if the equations are true.

Conclusion

In this article, we have discussed the concept of a system of linear equations and how to write an equivalent equation without denominators. We have also discussed the elimination method and how to use it to solve a system of linear equations. Additionally, we have answered some frequently asked questions about solving a system of linear equations.

Introduction

Writing an Equivalent Equation without Denominators

$\frac{x}{2} + \frac{y}{3} = 6$

Multiply both sides by 6:

$3x + 2y = 36$

Now, we have an equivalent equation without denominators.

Solving the System of Linear Equations

We have two equations:

$3x + 2y = 36$ ... (1)

$x - y = 2$ ... (2)

We can solve this system of linear equations using the method of substitution or elimination. In this case, we will use the elimination method.

Step 1: Multiply Equation (2) by 2

To eliminate the variable y, we need to multiply equation (2) by 2.

$x - y = 2$

Multiply both sides by 2:

$2x - 2y = 4$

Step 2: Add Equation (1) and the Resulting Equation

Now, we can add equation (1) and the resulting equation to eliminate the variable y.

$3x + 2y = 36$

$2x - 2y = 4$

Add both equations:

$5x = 40$

Step 3: Solve for x

Now, we can solve for x by dividing both sides of the equation by 5.

$5x = 40$

Divide both sides by 5:

$x = 8$

Step 4: Substitute x into Equation (2)

Now that we have the value of x, we can substitute it into equation (2) to solve for y.

$x - y = 2$

Substitute x = 8:

$8 - y = 2$

Step 5: Solve for y

Now, we can solve for y by subtracting 8 from both sides of the equation.

$8 - y = 2$

Subtract 8 from both sides:

$-y = -6$

Multiply both sides by -1:

$y = 6$

Conclusion

Introduction

Writing an Equivalent Equation without Denominators

$\frac{x}{2} + \frac{y}{3} = 6$

Multiply both sides by 6:

$3x + 2y = 36$

Now, we have an equivalent equation without denominators.

Solving the System of Linear Equations

We have two equations:

$3x + 2y = 36$ ... (1)

$x - y = 2$ ... (2)

We can solve this system of linear equations using the method of substitution or elimination. In this case, we will use the elimination method.

Step 1: Multiply Equation (2) by 2

To eliminate the variable y, we need to multiply equation (2) by 2.

$x - y = 2$

Multiply both sides by 2:

$2x - 2y = 4$

Step 2: Add Equation (1) and the Resulting Equation

Now, we can add equation (1) and the resulting equation to eliminate the variable y.

$3x + 2y = 36$

$2x - 2y = 4$

Add both equations:

$5x = 40$

Step 3: Solve for x

Now, we can solve for x by dividing both sides of the equation by 5.

$5x = 40$

Divide both sides by 5:

$x = 8$

Step 4: Substitute x into Equation (2)

Now that we have the value of x, we can substitute it into equation (2) to solve for y.

$x - y = 2$

Substitute x = 8:

$8 - y = 2$

Step 5: Solve for y

Now, we can solve for y by subtracting 8 from both sides of the equation.

$8 - y = 2$

Subtract 8 from both sides:

$-y = -6$

Multiply both sides by -1:

$y = 6$