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Chapter 2: Problem 95

Solve. Label any contradictions or identities. $$8.43 x-2.5(3.2-0.7 x)=-3.455 x+9.04$$

Short Answer

Expert verified
x = 1.25

Step by step solution

01

- Distribute the terms

First, distribute the -2.5distribute \text{over} the terms inside the parentheses:$$8.43x - 2.5(3.2 - 0.7x) = 8.43x - (2.5 \times 3.2) + (2.5 \times 0.7x)$$
02

- Simplify the expressions

Calculate the products:$$2.5 \times 3.2 = 8$$ and: $$2.5 \times 0.7x = 1.75x$$. The equation simplifies to: $$8.43x - 8 + 1.75x = -3.455x + 9.04$$
03

- Combine like terms

Combine the like terms on the left side:$$8.43x + 1.75x - 8 = -3.455x + 9.04$$ which simplifies to:$$10.18x - 8 = -3.455x + 9.04$$
04

- Move variables to one side

Add 3.455x to both sides of the equation to move the variables to one side: $$10.18x + 3.455x - 8 = 9.04$$ which simplifies to: $$13.635x - 8 = 9.04$$
05

- Isolate the variable term

Add 8 to both sides of the equation to isolate the variable term: $$13.635x - 8 + 8 = 9.04 + 8$$ which simplifies to: $$13.635x = 17.04$$
06

- Solve for x

Divide both sides by 13.635 to solve for x: $$x = \frac{17.04}{13.635}$$ Finally calculating the value gives us: $$x = 1.25$$

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Key Concepts

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

Distributive Property
The distributive property is a fundamental concept in algebra that allows you to multiply a single term across terms inside a parenthesis. It simplifies expressions and helps in solving equations.
For example, in our exercise: $$8.43x - 2.5(3.2 - 0.7x) = -3.455x + 9.04$$, we use the distributive property to multiply \text{-2.5} with both \text{3.2} and \text{0.7x}.
  • First, we calculate: $$-2.5 \times 3.2 = -8$$,
  • Then, we compute: $$-2.5 \times -0.7x = 1.75x$$.
This step helps us to rewrite the equation without parentheses, making it easier to combine and solve terms later.
Combining Like Terms
Combining like terms simplifies an equation and reduces it to a form where we can solve it. 'Like terms' are terms that have the same variable raised to the same power.
In our example: $$8.43x - 8 + 1.75x = -3.455x + 9.04$$, we focus on combining the terms that involve the variable x on the left side.
  • First, add the coefficients of x: $$8.43x + 1.75x = 10.18x$$.
Now, the equation reduces to: $$10.18x - 8 = -3.455x + 9.04$$, making it easier to isolate the variable.
Solving Linear Equations
Solving linear equations involves isolating the variable to find its value. This means making the variable term stand alone on one side of the equation.
In our exercise: $$10.18x - 8 = -3.455x + 9.04$$, we aim to move all x terms to one side and constants to the other.
  • Add \text{3.455x} to both sides: $$10.18x + 3.455x - 8 = 9.04$$, simplifying to: $$13.635x - 8 = 9.04$$.

  • Next, add 8 to both sides: $$13.635x - 8 + 8 = 9.04 + 8$$, which simplifies to: $$13.635x = 17.04$$.
Finally, we are ready to isolate the variable.
Isolating the Variable
Isolating the variable means getting the variable by itself on one side of the equation. This step gives us the solution to the equation.
Given: $$13.635x = 17.04$$, we isolate x by dividing both sides by \text{13.635}.
  • Calculate: $$x = \frac{17.04}{13.635}$$.

  • This results in: $$x = 1.25$$.
Therefore, the solution to our equation is: $$x = 1.25$$. Once the variable is isolated, we can see the value it represents. Understanding each step is crucial for solving similar problems.

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