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Chapter 2: Problem 62

Solve each equation, and check your solution. $$ (8 r-3)-(7 r+1)=-6 $$

Short Answer

Expert verified
r = -2

Step by step solution

01

Simplify the equation

Combine the like terms on the left-hand side of the equation. (8r - 3) - (7r + 1) = -6 8r - 3 - 7r - 1 = -6 Combine like terms: (8r - 7r) + (-3 - 1) = -6 r - 4 = -6
02

Solve for r

Isolate r by adding 4 to both sides of the equation: r - 4 + 4 = -6 + 4 r = -2
03

Check the solution

Substitute r = -2 back into the original equation to verify the solution is correct: (8(-2) - 3) - (7(-2) + 1) = -6 (-16 - 3) - (-14 + 1) = -6 (-19) - (-13) = -6 -19 + 13 = -6 -6 = -6 Since both sides are equal, the solution r = -2 is correct.

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Key Concepts

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

Combining Like Terms
Let's start with combining like terms. In algebra, like terms are terms that have the same variables raised to the same power. For example, in the equation ef gre Reco8r- 3 ) - ( 7r+ 1) = -6, you can see both 8r and 7r are like terms because they both have the variable r. By combining these terms, we directly simplify the equation. Here's how you do it:

8r - 3 - 7r - 1 = -6
- First, distribute the negative sign and rewrite the expression inside the parentheses.
Now, combine the like terms:
(8r - 7r) + (-3 - 1) = -6 .

This gives you r - 4 = -6 .

By focusing on like terms and simplifying them, you make the equation easier to solve.
Isolating the Variable
Next, we want to isolate the variable. This means getting the variable alone on one side of the equation, which allows us to solve for it.

Continuing from where we left off: r - 4 = -6, we need to move the -4 to the other side.
  • To do this, add 4 to both sides of the equation:
r - 4 + 4 = -6 + 4 .
This simplifies to r = -2 .

When you isolate the variable, think of balancing the equation: whatever you do on one side, you must also do on the other side to keep everything equal.
Checking Solutions
Finally, let's check our solution to ensure it is correct. Checking solutions is an important step in solving equations as it verifies our answer.
  • To check the solution, substitute r = -2 back into the original equation:
(8(-2) - 3) - (7(-2) + 1) = -6.
Simplify within the parentheses: ( -16 - 3 ) - ( -14 + 1 ) . This results in -19 - ( -13 ) .

Since subtracting a negative is the same as adding a positive, we get -19 + 13 = -6 .

Both sides of the equation are equal, confirming that our solution r = -2 is indeed correct.

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