Question

MENTAL MATH Write a question that could be used to solve the equation. Then use mental math to solve the equation. $$4 p=36$$

Short answer

The solution to the equation \(4p=36\) is \(p = 9\).

Step-by-step solution

Identify Operations to Isolate \(p\)

Notice that \(p\) is currently multiplied by 4 in the equation \(4p = 36\). To isolate \(p\), you want to undo that multiplication by performing the opposite operation, which is division.

Perform the Operation

Divide both sides of the equation by 4. \(4p ÷ 4 = 36 ÷ 4\) results in the new equation \(p = 9\).

Check the Solution

Substitute \(p = 9\) back into the original equation to check if it holds true. \(4*9 = 36\), so indeed, our solution is correct.

Key concepts

Isolate the Variable

Isolating the variable in an equation is essential when it comes to solving for that variable. This process involves manipulating the equation in such a way that the variable you're solving for stands alone on one side of the equation. To isolate the variable in the equation \(4p = 36\), we focus on what's happening to the variable and, more importantly, how we can reverse that operation.
For example, if the variable is being multiplied by a number, as it is in this equation with the number 4, we must do the opposite to free it. Isolating the variable makes the solution clear and understandable, as it lets the student see exactly what the variable equals without other numbers or operations interfering. Once isolated, as in \(p = 9\), it becomes a simple matter to understand and verify the solution.

Opposite Operations

Opposite operations, also known as inverse operations, are the backbones of solving algebraic equations. They are operations that reverse the effect of each other, like addition and subtraction, or multiplication and division. When looking at the equation \(4p = 36\), we identify multiplication as the operation affecting the variable \(p\).
To 'undo' the multiplication and solve for \(p\), we perform the opposite operation – in this case, division. Therefore, we divide both sides of the equation by 4, which simplifies to \(p = 9\). When using opposite operations, it is crucial that what we do to one side of the equation, we also do to the other side to maintain balance. This idea is rooted in the fundamental property of equality, ensuring that the two sides remain equal after the operation.

Check the Solution

After solving an equation, it's important to verify that the solution is correct. This last step, known as checking the solution, is done by substituting the found value back into the original equation. For the equation \(4p = 36\), after deducing that \(p = 9\), we check the solution by plugging in 9 for \(p\) in the original equation to see if the equality holds true.
Thus, we compute \(4 \times 9\) and verify that it equals 36. Since it does, we can confidently say that the solution \(p = 9\) is correct. Skipping this step could lead to unnoticed mistakes, so students are encouraged to always check their work to reinforce their understanding and ensure accuracy.