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Chapter 2: Problem 77

Solve each equation, if possible. $$ -3 p+7=-3 $$

Short Answer

Expert verified
\(p = \frac{10}{3}\)

Step by step solution

01

Isolate the variable term

Start by moving the constant term on the left side of the equation to the right side by subtracting 7 from both sides: \(-3p + 7 - 7 = -3 - 7\).This simplifies to:\(-3p = -10\).
02

Solve for the variable

To solve for \(p\), divide both sides of the equation by -3:\(p = \frac{-10}{-3}\).This simplifies to:\(p = \frac{10}{3}\).

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Key Concepts

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

Isolating the Variable
When solving linear equations, isolating the variable is often the first step. It means rearranging the equation so that the variable is alone on one side of the equation. This is crucial because isolating the variable allows us to find its value more easily.
  • Identify what you need to do to both sides of the equation to have the variable alone.
  • Perform operations to both sides to maintain the balance of the equation.
  • Use inverse operations to remove coefficients or terms associated with the variable.
This process simplifies the equation to a point where you can clearly solve for the variable. Let's consider the expression \(-3p + 7 = -3\). To isolate\(p\), the constant term, +7, needs to be moved to the other side, which we do by subtracting 7 from both sides, resulting in\(-3p = -10\). There you have it 鈥� the variable\(p\) is isolated.
Constant Term
A constant term in an equation does not contain the variable we are trying to solve for. It stands by itself as a fixed number without a variable attached. Identifying and working with constant terms is an essential part of solving linear equations.
  • The constant term is often added or subtracted from both sides to aid in isolating the variable.
  • In our equation, \(-3p + 7 = -3\), the number 7 is the constant term on the left side.
  • The operation needed here was subtraction, to eliminate the +7 effectively from the equation.
Removing the constant term adjusts the equation properly, paving the way for other necessary operations. By doing this correctly, as seen with 7 being subtracted from both sides to yield\(-3p = -10\), the equation becomes much simpler to solve.
Division in Algebra
Once the variable is isolated with terms grouped correctly, sometimes we need to use division to solve for the variable itself. Division is applied especially when the variable is multiplied by a coefficient.If we have\(-3p = -10\), to solve for\(p\), we must divide both sides by -3, the coefficient in front of\(p\). Here is how:
  • Divide every term by the coefficient of the variable.
  • Both sides should be divided so the equation remains balanced.
  • In this example, dividing -10 by -3 gives us\(p = \frac{10}{3}\).
This technique allows you to solve for the variable by reducing the equation to its simplest form. It鈥檚 helpful to remember that dividing by a negative turns negative values into positive, making the result for\(p\) in this example a positive\(\frac{10}{3}\).

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