Solving Linear Equations: A Step-by-Step Guide with Practical Examples
Introduction
Understanding how to solve linear equations is a fundamental skill in algebra. Linear equations are equations of the first degree, meaning that the variable appears only to the first power. This article will walk you through the process of solving a specific equation, 3x - 4 -24 - x, and provide a general algorithm for solving such equations.
Step-by-Step Solution of 3x - 4 -24 - x
Let's begin by solving the equation: 3x - 4 -24 - x.
Step 1: Eliminate the Brackets
There are no brackets to eliminate in this case, but if there were, we would distribute the terms outside the brackets to the terms inside. For example, if we had 3(x - 4) -24 - x, we would need to distribute the 3 to both terms inside the bracket.
Step 2: Combine Like Terms
Our equation simplifies directly to: 3x - 4 -24 - x.
Next, we will combine the terms involving x on one side and the constant terms on the other side. To do this, we need to add x to both sides of the equation, and add 4 to both sides of the equation.
3x x - 4 -24 - x x 4
Simplifying, we get: 4x - 4 -24 4
4x - 4 -20
Step 3: Isolate the Variable Term
To isolate the term with the variable, we need to move the constant terms to the right side. We can do this by adding 4 to both sides of the equation.
4x - 4 4 -20 44x -16
Step 4: Solve for x
The final step is to isolate the variable x by dividing both sides by 4.
4x / 4 -16 / 4x -4
Therefore, the value of x is -4.
General Algorithm for Solving Linear Equations
Let's now look at a more generalized approach. The following algorithm can be used to solve any linear equation of the form ax b cx d.
Step 1: Group the x Terms
Move all the terms with x to one side of the equation and all the constant terms to the other side. To do this, subtract cx from both sides and subtract b from both sides.
ax - cx d - b(a - c)x d - b
Step 2: Isolate the Variable
Divide both sides of the equation by (a - c) to solve for x.
x (d - b) / (a - c)This gives you the value of x in a general form.
Alternative Methods and Examples
Let's look at other ways to solve the equation 3x - 4 -24 - x and see if we get the same result.
Example 1
Delete brackets (if any) and simplify the equation:
3x - 4 -24 - x3x - 4 -24 - x
3x - 4 -24 - x
Add 2x to both sides to combine the x terms on one side: 3x - 4 2x -24 - x 2x
5x - 4 -24
Add 4 to both sides to isolate the term with x: 5x - 4 4 -24 4
5x -20
Divide both sides by 5 to solve for x: x -20 / 5 -4
Example 2
Subtract 2x from both sides to start isolating x:
3x - 2x - 4 -24 - x - 2xx - 4 -24 - 3x
Add 4 to both sides to isolate the term with x: x - 4 4 -24 - 3x 4
x -20 - 3x
Add 3x to both sides to combine the x terms on one side: x 3x -20
4x -20
Divide both sides by 4 to solve for x: x -20 / 4 -5
Note that this example is incorrect, as the correct solution is x -4.
Conclusion
Solving linear equations is a crucial skill in algebra. By following a systematic approach, such as the one outlined in this article, you can confidently solve any linear equation and ensure that your answer is accurate. The general algorithm applied here can be adapted to solve a wide range of similar equations.