Question

Solve the equation. Check your solution. \(\frac{9}{10} x=0.18\)

Short answer

The solution to the equation is \(x = 0.2\).

Step-by-step solution

Identify the equation

The equation given is \(\frac{9}{10} x = 0.18\). This is a simple linear equation where \(x\) needs to be isolated.

Isolate x

To isolate \(x\), divide both sides of the equation by \(\frac{9}{10}\), getting \(x = \frac{0.18}{\frac{9}{10}}\)

Simplify the numerical value

Carry out the division to obtain \(x = 0.2\)

Check the solution

Substitute \(x = 0.2\) into the original equation \(\frac{9}{10} x = 0.18\) to verify if the left-hand side equals the right-hand side. It should resolve to \(0.18 = 0.18\), confirming the solution is correct.

Key concepts

Solving Equations

Solving equations is like solving a puzzle. The aim is to find the value of the variable that makes the equation true. In the case of our exercise, we need to find the value of \( x \) in the equation \( \frac{9}{10} x = 0.18 \).
To solve, you look at the equation and decide what operation will isolate the variable \( x \). Each operation done to one side of the equation must also be applied to the other. This keeps the equation balanced, just like a seesaw.
Here, we have a fraction multiplying \( x \). To solve it, we multiply both sides by the reciprocal of the fraction. This undoes the multiplication and isolates \( x \).
Solving equations requires careful attention to details, ensuring each step follows logically from the last. It's similar to following a recipe: each step builds on the previous one until you have the finished dish.

Isolating Variables

Isolating the variable is all about getting \( x \) on its own, standing like a lighthouse on an ocean. In our equation \( \frac{9}{10} x = 0.18 \), isolating \( x \) means removing the fraction \( \frac{9}{10} \) from it.
To do this, we multiply both sides of the equation by the reciprocal of \( \frac{9}{10} \), which is \( \frac{10}{9} \).

• Multiply: \( \frac{10}{9} \times \frac{9}{10} x = \frac{10}{9} \times 0.18 \)
• The fractions on the left cancel out, leaving \( x \)
• Perform the multiplication on the right to find \( x \)

When the variable is isolated, it's like untangling a knot.
The value of \( x \) then stands alone, and you can see it clearly!

Checking Solutions

After solving for the variable, it's crucial to check your work. Checking solutions ensures that all calculations were correct and the value found truly solves the original equation.
Here's how you check your solution:

• Take the value of \( x \) found, which is 0.2 in this problem
• Substitute it back into the original equation \( \frac{9}{10} x = 0.18 \)
• Replace \( x \) with 0.2: \( \frac{9}{10} \times 0.2 \)
• Simplify to confirm: \( 0.18 = 0.18 \)

By seeing both sides of the equation equal, you can confidently say the solution is correct.
It's a satisfying confirmation that all your hard work has paid off!