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Chapter 2: Problem 203

Solve the following equations with variables and constants on both sides. $$12 q-5=9 q-20$$

Short Answer

Expert verified
q = -5

Step by step solution

01

Isolate the variable term

Begin by moving all terms involving the variable to one side of the equation. Subtract \(9q\) from both sides:\[12q - 9q - 5 = 9q - 9q - 20\]Simplifies to:\[3q - 5 = -20\]
02

Isolate the constant term

Next, move the constant term on the left side to the right side by adding 5 to both sides:\[3q - 5 + 5 = -20 + 5\]This simplifies to:\[3q = -15\]
03

Solve for the variable

Finally, solve for \(q\) by dividing both sides by 3:\[q = \frac{-15}{3}\]This simplifies to:\[q = -5\]

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Key Concepts

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

Solving Equations
When it comes to solving algebraic equations, it's all about finding the value of the variable that makes the equation true. An equation is like a balanced scale, and whatever you do to one side, you must also do to the other to maintain the balance.

In our exercise, we start with the equation '12q - 5 = 9q - 20' To solve it, we go through a series of steps to simplify and eventually find the value of the variable. These steps involve moving terms involving the variable to one side and constants to the other. Always remember:
* Keep the equation balanced.
* Combine like terms.

By following these principles, you'll find the right value for your variable.
Isolating Variables
Isolating the variable is a crucial skill in solving equations. It means getting the variable by itself on one side of the equation.
Here鈥檚 how we do it step by step:
  • Identify terms with the variable (e.g., '12q' and '9q').
In our exercise, we moved '9q' to the left side by subtracting '9q' from both sides:
'12q - 9q - 5 = 9q - 9q - 20' , which simplifies to '3q - 5 = -20'


Next, we isolate the variable by moving the constants to the other side:
  • Add or subtract the constant (e.g., +5 on both sides).
This changes our equation to:
  • '3q = -15'
Now, 'q' is getting closer to being isolated.
Basic Algebra
Understanding basic algebra is essential for solving more complex mathematical problems. Algebra involves working with unknowns (represented by variables) and numbers in an equation.
Let's break down the process of solving our example step by step:
In the equation '3q = -15', we have our variable 'q' multiplied by 3. To solve for 'q', we do the opposite of multiplication, which is division:
'q = -15 / 3'
This simplifies to:
'q = -5'
  • Always perform the same operation on both sides.
  • Keep practicing to get comfortable with variables and operations.

    Remember, algebra is like a puzzle. The more you practice, the better you'll get at seeing the patterns and knowing which steps to take.

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Most popular questions from this chapter

Solve each inequality, graph the solution on the number line, and write the solution in interval notation. $$23 p-2(6-5 p)>3(11 p-4)$$

Graph each inequality on the number line and write in interval notation. (a) \(\quad x \geq-4\) (b) \(\quad x<2.5\) (c) \(x>-\frac{3}{2}\)

Solve each inequality, graph the solution on the number line, and write the solution in interval notation. $$6 n-12(3-n) \leq 9(n-4)+9 n$$

Solve each inequality, graph the solution on the number line, and write the solution in interval notation. $$u+25>21$$

Translate and solve. Then write the solution in interval notation and graph on the number line. Nineteen less than \(b\) is at most -22.
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