Problem 30 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ 4+6 x-9 x=3 x $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 2/3\).

Step 1: Simplify Both Sides

First of all, simplify the numbers with the same variable by adding or subtracting them from each other. The equation simplifies to \(-3x + 4 = 3x\) on simplifying \(6x - 9x\) and \(3x\).

Step 2: Isolate x Term

Now, isolate the variable \(x\) on one side, add \(3x\) to both sides of the equation results in: \(4 = 6x\).

Step 3: Solve for x

Finally, calculate the value of \(x\) by dividing both sides of the equation by 6: \(x = 4/6\).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (like the letter x), and arithmetic operations such as addition, subtraction, multiplication, and division. In the given exercise, 4 + 6x - 9x = 3x, we see a simple algebraic expression that needs to be solved for the variable x.

Primarily, algebraic expressions require simplification by combining like terms, which are terms that contain the same variable to the same power. In our case, 6x and -9x are like terms and can be combined by subtraction to simplify the expression on the left side of the equation.

Isolating the Variable

To find the value of a variable, we need to 'isolate' it on one side of the equation. This means that we want to have the variable by itself on one side, usually the left or right, and the numbers or constants on the other side.

As seen in the solution, after combining like terms, one way to isolate x is by adding or subtracting terms on both sides of the equation to get the variable on one side and the constants on the other. This is demonstrated by adding 3x to both sides to eliminate it from the right and bring it to the left, resulting in a simplified form of 4 = 6x.

Equation Simplification

Equation simplification involves reducing an equation to its simplest form to make it easier to solve. It means carrying out basic arithmetic operations and combining like terms, when possible, to get a clearer picture of what you're working with.

In our example, once we have 4 = 6x, simplifying further to solve for x is straightforward. Since x is being multiplied by 6, we do the opposite operation—division—on both sides, to isolate x. Dividing 4 by 6 gives us the final value of x. Remember, keeping track of your operations is crucial to maintaining the integrity of the equation, as the same operation must be applied to both sides to keep the equation balanced.