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Chapter 3: Problem 30

Solve the equation. $$4-2 p=-3+5 p$$

Short Answer

Expert verified
p = 1

Step by step solution

01

Combine like terms

First, we'll get all terms containing the variable on one side of the equation and constant terms on the other side. To do this, add 2p to both sides and add 3 to both sides of the equation.
02

Simplify the equation

After adding, the equation becomes \[4 + 3 = 2p + 5p\], which simplifies to \[7 = 7p\]
03

Solve for the variable

Next, we solve for p by dividing both sides by 7. \[p = \frac{7}{7} = 1\]

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Key Concepts

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

Combining Like Terms
Combining like terms is a crucial step in solving linear equations. It helps to simplify the equation and make it easier to solve. In our original exercise, we start with the equation: $$4 - 2p = -3 + 5p$$.
Here鈥檚 how to identify and combine like terms:
  • 馃攳 Terms with the same variable part are like terms. For example, \textbf{-2p and 5p} are like terms because they both contain the variable \textbf{p}.
  • 馃攳 Constant terms, which do not contain any variables (like \textbf{4 and -3}), are also like terms with each other.
To combine like terms, we need to move them to one side of the equation. Add \textbf{2p} to both sides and then add \textbf{3} to both sides to get all variable terms on one side and constants on the other:
$$4 + 3 = 2p + 5p$$.
Simplifying Equations
After combining like terms, the next step is simplifying the equation. This process makes the equation much easier to solve. We look at the equation after combining like terms: $$4 + 3 = 2p + 5p$$.
Simplify both sides of the equation by:
  • 馃攳Adding the constants on the left side: \textbf{4 + 3 = 7}
  • 馃攳Adding the coefficients of the variable terms on the right: \textbf{2p + 5p = 7p}
By simplifying, we get a more straightforward equation: $$7 = 7p$$.
Now, the equation is in a simplified form, making it much easier to solve.
Solving for Variables
The last step in solving linear equations is to isolate the variable. This means getting the variable on one side of the equation and everything else on the other side. From our simplified equation: $$7 = 7p$$,
we need to solve for \textbf{p}.
  • 馃攳 To isolate \textbf{p}, we divide both sides of the equation by 7: \textbf{\textbackslash[ p = \frac{7}{7} \textbackslash]}.
  • 馃攳 This leaves us with \textbf{p = 1}.
Now we have solved for the variable, which means \textbf{p} equals 1.
By following these steps: combining like terms, simplifying the equation, and solving for the variable, we can efficiently tackle any linear equation. Each step is critical in breaking down the problem into more manageable parts.

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