Solving Linear Equations: A Practical Guide

Linear equations are a fundamental component of algebra and play a crucial role in many real-life applications such as finance, physics, and engineering. In this guide, we will explore how to solve for y when x is given a specific value, using the example -2x - 3y 6 when x -1.

Understanding the Substitution Method

The goal of solving for y, given that x has a specific value, is to isolate y on one side of the equation. The substitution method involves substituting the given value of x into the equation and then simplifying it step by step.

Step-by-Step Solution

Let's start with the given equation and the value of x:

Given: -2x - 3y 6, and x -1

Substitute -1 for x in the equation:

-2(-1) - 3y 6

When substituting, remember to pay attention to the signs:

-2(-1) 2

So the equation becomes: 2 - 3y 6

Next, we isolate y by moving the constant term to the other side:

Subtract 2 from both sides of the equation:

2 - 2 - 3y 6 - 2

This simplifies to:

-3y 4

Finally, we solve for y by dividing both sides by -3:

[ frac{-3y}{-3} frac{4}{-3} ]

[ y -frac{4}{3} ]

Thus, the solution is (y -frac{4}{3}).

Verification

To verify the solution, we substitute the found value of y back into the original equation and check if it satisfies the equation:

-2(-1) - 3left(-frac{4}{3}right) 6

2 4 6

6 6

The solution is correct since the left-hand side equals the right-hand side.

Applying the Substitution Method

The substitution method can be applied to solve various linear equations. Here are a couple of additional examples to solidify your understanding:

Example 1: Solve for y when x 2 in the equation 7x - 5y 14

Substitute x 2 into the equation:

7(2) - 5y 14

Subtract 14 from both sides:

14 - 14 - 5y 14 - 14

-5y 0

Solve for y by dividing both sides by -5:

[ frac{-5y}{-5} frac{0}{-5} ] ]

[ y 0 ] ]

Example 2: Solve for y when x -3 in the equation 3x 2y -12

Substitute x -3 into the equation:

3(-3) 2y -12

Subtract 9 from both sides:

-9 9 2y -12 9

2y -3

Solve for y by dividing both sides by 2:

[ frac{2y}{2} frac{-3}{2} ] ]

[ y -frac{3}{2} ] ]

By following these steps, you can systematically solve for y given any value of x and any linear equation.

Understanding the substitution method and practicing similar problems will greatly enhance your algebraic skills and problem-solving abilities.