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Chapter 3: Problem 54

Solve and check each equation. $$-6=9-3 p$$

Short Answer

Expert verified
The solution to the equation \(-6 = 9 - 3p\) is \(p = 5\). The solution is verified by substituting the value of p back into the original equation.

Step by step solution

01

Write down the given equation

The given equation is: \(-6 = 9 - 3p\)
02

Add 3p to both sides

By adding 3p to both sides, we can make the variable term on the left-hand side: \(-6 + 3p = 9 - 3p + 3p\) Simplifying, we get: \(3p - 6 = 9\)
03

Add 6 to both sides

Now, we can isolate p further by adding 6 to both sides of the equation: \(3p - 6 + 6 = 9 + 6\) Simplifying, we get: \(3p = 15\)
04

Divide both sides by 3

To find the value of p, simply divide both sides of the equation by 3: \(\frac{3p}{3} = \frac{15}{3}\) Simplifying, we get: \(p = 5\)
05

Check the solution

To check if our solution is correct, plug the value of p back into the original equation: \(-6 = 9 - 3(5)\) \(-6 = 9 - 15\) \(-6 = -6\) Since both sides of the equation are equal, the solution is correct. Therefore, the value of p is 5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Equations
To solve an equation means finding the value of the variable that makes the equation true. In our case, the equation is \(-6 = 9 - 3p\). The goal is to isolate the variable \(p\) on one side of the equation, usually the left. This involves performing operations that keep the equation balanced, much like keeping a scale even on both sides.

Here's how you can systematically approach this problem:
  • Address each side of the equation with an eye for symmetry. Any operation you apply on one side, do it on the other too.
  • Start by eliminating constants on the side with the variable, making the coefficient manageable.
  • Use operations in the reverse order of BIDMAS/BODMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction) when solving for a variable.
By following these principles, you will ensure that you maintain the equality of the equation throughout your calculations.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying equations to make them easier to solve. In the equation \(-6 = 9 - 3p\), the goal is to isolate \(p\). This requires some key manipulation steps.

First, we added \(3p\) to both sides of the equation: \(-6 + 3p = 9 - 3p + 3p\). By doing this, the \(3p\) terms on the right side cancel each other, simplifying the equation to \(3p - 6 = 9\).
  • The next step involved adding 6 on both sides: \(3p - 6 + 6 = 9 + 6\). This isolates the \(3p\) term on the left.
  • Finally, we divide by the coefficient of \(p\), which is 3: \(\frac{3p}{3} = \frac{15}{3}\), resulting in \(p = 5\).
These steps simplify the equation, making it easier to solve, showing how effective algebraic manipulation can be in problem-solving.
Verification of Solutions
Verification is crucial to ensure the value found for the variable truly satisfies the original equation. In this exercise, we've found \(p = 5\) after solving the equation, but we must check if this is correct.

We substitute \(p = 5\) back into the original equation and simplify:
  • Original equation: \(-6 = 9 - 3p\)
  • Substitute \(p = 5\): \(-6 = 9 - 3(5)\)
  • Compute: \(-6 = 9 - 15\)
  • Simplify: \(-6 = -6\)
The left side and the right side of the equation are equal, confirming our solution is correct. This process of substitution and simplification not only validates our answer but also provides assurance of its accuracy. Verification is an essential step in algebra to affirm the solution confidently.

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