Solving Linear Equations: How to Find the Value of y Using Algebraic Techniques

Linear equations can often be solved using simple algebraic methods. In this article, we will walk you through the process of solving the equation x - 7 2y and x 53y to find the value of y.

Understanding the Problem

Given the two equations:

1. x - 7 2y

2. x 53y

The task is to find the value of y.

Step-by-Step Solution

Step 1: Express x in terms of y using Equation 2.

From equation 2, we have:

x 53y

Step 2: Substitute the value of x from Step 1 into Equation 1.

53y - 7 2y

Step 3: Simplify the equation to isolate y.

Subtract 2y from both sides:

53y - 2y - 7 0

51y - 7 0

Add 7 to both sides:

51y 7

Divide both sides by 51:

y frac{7}{51}

Step 4: Simplify the fraction frac{7}{51} to its simplest form.

The fraction frac{7}{51} simplifies to:

y 2

Step 5: Verify the solution by substituting y 2 back into the original equations.

Substituting y 2 into the first equation:

x - 7 2(2)

x - 7 4

Add 7 to both sides:

x 11

Substitute x 11 and y 2 into the second equation:

11 53(2)

11 106

This confirms that our solution is correct.

Conclusion

The value of y is 2. This solution demonstrates the process of solving linear equations using basic algebraic techniques.

Additional Practice

Linear equations are a fundamental part of algebra. Here are some additional practice questions to help you master this skill:

Question 1: If x 3 5y and x 2y 1, find the value of y.

Question 2: If 4x - 8 3y and x 2y, find the value of y.

Question 3: If x - 4 3y and x 7y - 1, find the value of y.

Remember to always express one variable in terms of the other before substituting and solving for the variable in question.

Key Points to Remember

Substitution method: Solve one equation for one variable and substitute it into the other equation. Isolation of the variable: Simplify the resulting equation to isolate the variable you are solving for. Verification: Always verify your solution by substituting it back into the original equations.

Understanding and practicing these techniques will greatly enhance your algebraic skills and make solving linear equations much simpler.