Problem 49 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. $$2 x=-3 x$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 0\).

Step 1: Simplify the equation

Let's simplify the equation by bringing similar terms together. This can be done by adding \(3x\) to both sides to get \(2x + 3x = 0\). Further simplifying this gives \(5x = 0\).

Step 2: Solve for x

Now let's solve for 'x' by dividing both sides by 5. This leads to \(x = 0\).

Step 3: Check the solution

Replacing 'x' in the original equation with the obtained value we have: \(2(0)=-3(0)\), which checks out because both sides of the equation equal 0.

Key Concepts

Simplify Equations

When faced with algebraic equations, the first and quintessential step is to simplify the equation. Simplification often involves combining like terms, which are terms that contain the same variable raised to the same power. For instance, in our example of solving the equation 2x = -3x, we identify that on either side of the equal sign, there are terms involving the variable 'x'. By adding 3x to both sides, we effectively consolidate these like terms which simplifies the equation to 5x = 0.

This process makes the equation more manageable and sets the stage for the next step which is finding the value of the variable. It's also a good practice to eliminate any unnecessary terms or factors and ensure the equation is as straightforward as possible before proceeding to solve for the variable.

Solve for Variable

Once the equation is simplified, the next goal is to isolate the variable you are solving for, in this case 'x'. Isolation means to get the variable on one side of the equation, ideally with a coefficient of 1. In the simplified equation 5x = 0, 'x' is already by itself on one side, but it's multiplied by 5. To isolate 'x', divide both sides of the equation by 5, because division is the inverse operation of multiplication.

Performing this operation correctly gives us x = 0, which signifies that 'x' can only be zero for this particular equation. Isolating the variable often involves reverse operations; multiplication and division for additive and subtractive terms, and vice versa. Always perform the same operation on both sides of the equation to maintain balance.

Check Solution in Algebra

After solving for the variable, it is crucial to verify that the solution is correct. This involves substituting the value back into the original equation and checking if both sides are equivalent. For our example where the solution found was x = 0, we substitute 'x' with 0 in the original equation 2x = -3x. After replacing 'x' we get 2(0) = -3(0), this simplifies to 0 = 0, which is a true statement.

Therefore, our solution is correct. This step is important not just to confirm the correctness of your answer but also to understand the importance of balance in an equation. If at any time the solution does not check out, it means there was a mistake in the earlier steps and you should revisit the equation to find and correct the error.