Problem 67 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. $$10 x=50$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 5\)

Step 1: Isolate the variable x

To isolate the variable x, divide both sides of the equation by the coefficient of x, which is 10. The equation therefore becomes \(x = 50/10\)

Step 2: Simplify the right side of the equation

After the division, simplify the right side of the equation to obtain the solution for x. This yields \(x = 5\)

Step 3: Check the solution

Substitute the found value for x back into the original equation to check if the equation holds true. \(10*5 = 50\). As we can see, the substitution yields the same value on both sides of the equation, confirming that the solution is correct.

Key Concepts

Isolate the variable

When solving linear equations, the first trick is to isolate the variable, which is the unknown value we are looking for. Think of it as zooming in on the "x" in this case, and getting it by itself. In order to do this, we want to

• Eliminate anything that is multiplied, added, or subtracted to the variable.
• Perform the inverse operation to both sides of the equation.

For the equation given, \(10x = 50\), the variable "x" is multiplied by 10. To isolate "x", we do the opposite of multiplication, which is division.

In this case, you divide both sides of the equation by 10, leading to \(x = \frac{50}{10}\). Now "x" is alone on one side, making way for the next step.

Equation simplification

Once the variable is isolated, the next step is to simplify the equation to find out exactly what the variable equals. Simplification involves performing the arithmetic indicated by the equation after isolating the variable.For our example, \(x = \frac{50}{10}\), the division \(50 \div 10\) needs to be calculated. We simplify it to determine that \(x = 5\).

Simplifying equations is all about completing these basic math operations step by step, such as division or multiplication, ensuring that we follow the operations until the variable is clearly defined.Ensuring clarity in each step makes it much easier to keep track of what "x" equals, which is 5 in this case.

Check the solution

The final step in solving an equation is checking the solution to ensure correctness. This verification process assures us that the value found for the variable indeed satisfies the original equation.To check, replace the variable in the original equation with the solution found. For \(10x = 50\), substitute "x" with 5, which gives us \(10 imes 5 = 50\).

Since both sides of the equation equal 50, it confirms that the solution is correct. Always perform this step because it confirms your calculations and ensures no errors were made in the preceding steps. Verification acts as a final checkmark to conclude that the solution is true and reliable for the problem given.