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Chapter 11: Problem 63

Solve. $$ a+(a-3)=(a+2)-(a+1) $$

Short Answer

Expert verified
a = 2

Step by step solution

01

Simplify both sides

Simplify the expression on both sides of the equation. The left side is already simplified to a + (a-3)= 2a - 3.Now simplify the right side:(a+2) - (a+1)= a + 2 - a - 1= 1.
02

Equate the simplified expressions

Write the simplified expressions from Step 1 as an equation:2a - 3 = 1.
03

Solve for a

Solve the equation for a. First, add 3 to both sides:2a - 3 + 3 = 1 + 32a = 4.Then, divide both sides by 2:a = 2.

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Key Concepts

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

simplifying expressions
Simplifying expressions means making a math expression shorter or more manageable. Given our original exercise
that starts with:
$$ a + (a-3) = (a+2)-(a+1) $$
the goal is to remove parentheses and combine like terms.
Begin with the left side:
  • Distribute the operation within parentheses:
    $$ a + (a-3) = a + a - 3 $$
  • Combine like terms, which are the 'a' terms in this case:
    $$ 2a - 3 $$
Now, simplify the right side:
  • Distribute the operation within parentheses:
    $$ (a+2) - (a+1) = a + 2 - a - 1 $$
  • Combine like terms by canceling out 'a' terms and adding constants:
    $$ 1 $$
Simplifying expressions makes solving equations easier, so always handle parentheses and combine like terms first.
equating expressions
After simplifying both sides of the equation, the next step is to set the simplified expressions equal to one another.
From our example:
$$ 2a - 3 = 1 $$
Here is how it works:
  • Take the simplified version of the left side of the equation:
    $$ 2a - 3 $$
  • Take the simplified version of the right side of the equation:
    $$ 1 $$
  • Write an equation that shows these two expressions are equal:
    $$ 2a - 3 = 1 $$
Equating expressions is essential because it allows us to form a single equation that can be solved to find the unknown variable.
solving for variables
The final step is solving for the variable. We aim to isolate the variable on one side of the equation.
Using our example:
$$ 2a - 3 = 1 $$
we'll solve for 'a' in two simple steps.
  • First, add 3 to both sides to move the constant term:
    $$ 2a - 3 + 3 = 1 + 3 $$

    Simplifies to:
    $$ 2a = 4 $$
  • Next, divide both sides by 2 to isolate the variable 'a':
    $$ \frac{2a}{2} = \frac{4}{2} $$

    Simplifies to:
    $$ a = 2 $$
Solving for the variable involves performing inverse operations to remove coefficients and constants, leaving the variable alone.

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