Problem 39 - Algebra 1

Question

Solve the equation $$ -4 a-3=6 a+2 $$

Step-by-Step Solution

Answer

The solution of the equation is \(a = -0.5\).

Step 1: Rearrange terms

Move all terms with \(a\) to one side (preferably left) and all constants to the other side (preferably right) to form an equation in the standard form. It gives \(-4a - 6a = 2 + 3\) or \(-10a = 5\).

Step 2: Finding the value of a

Now divide by -10 on both sides to solve for \(a\). This gives \(a = \frac{5}{-10}\).

Key Concepts

Rearranging Terms

In solving linear equations, a crucial initial step is rearranging terms. This means moving terms involving the variable, like those with the unknown "a" in this case, to one side of the equation, typically the left. Meanwhile, all constant terms, which are numbers without a variable, should go on the opposite side, preferably to the right. This separation is key to simplifying the equation into a form that is easier to solve.

In the equation \(-4a - 3 = 6a + 2\), adjust the positioning of terms:

• First, remove all instances of "a" from the right side by subtracting \(6a\) from both sides.
• Then shift the constant term \(-3\) to the right by adding \(3\) to both sides.

This leaves us with \(-4a - 6a = 2 + 3\). On simplification, it becomes \(-10a = 5\). Rearranging terms like this sets the stage for finding the unknown with relative ease.

Finding Variables

After rearranging the equation, the next step is to simplify and find the variable. Here, simplification leads us to a new equation, ideally one that isolates the terms with our variable. In the current scenario, we've arrived at:

• \(-10a = 5\)

Solving this involves simplifying the equation further. This often involves arithmetic operations such as addition, subtraction, multiplication, or division.

In practice, you can think of finding variables as peeling back layers to get to the core, or the truth, about what the variable equals. Notice how all terms not involving "a" have been eliminated from the left side, and those without a constant are on the right.

Breaking down this equation by simplifying leads to clearer insights into the variable's relationship with other numbers in the equation.

Solving for a Variable

The final step in handling linear equations is solving for the variable. After isolating terms with "a" and simplifying our equation to \(-10a = 5\), we aim to make the coefficient of "a" equal to 1. This is done by dividing every term by \(-10\). Performing this operation results in:

• \(a = \frac{5}{-10}\)

Further simplification gives the result:

• \(a = -\frac{1}{2}\)

This simple division transforms the equation such that the exact value of "a" is clear and standalone. Remember that solving for a variable essentially means isolating the variable on one side and determining its value. This process is pivotal in reaching the final solution, which is our ultimate objective. By completing these steps, we unravel the mystery of the original equation, transforming unknowns into known quantities.