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Chapter 2: Problem 45

For exercises 1-72, (a) solve. (b) check. $$ 0.2 x+1.86=5.44 $$

Short Answer

Expert verified
x = 17.9

Step by step solution

01

Isolate the Variable

Start by isolating the variable term on one side of the equation. Subtract 1.86 from both sides: \[0.2x + 1.86 - 1.86 = 5.44 - 1.86\]This simplifies to: \[0.2x = 3.58\]
02

Solve for x

Now isolate x by dividing both sides by 0.2:\[x = \frac{3.58}{0.2}\]This gives us:\[x = 17.9\]
03

Check the Solution

Substitute x back into the original equation to verify the solution:\[0.2(17.9) + 1.86 = 5.44\]Calculate the left side:\[3.58 + 1.86 = 5.44\]Since both sides are equal, the solution is verified.

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Key Concepts

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

Isolate the Variable
To solve a linear equation, our main goal is to find the value of the variable. However, the variable is often mixed with other numbers. In this example, we need to isolate the variable term (which is 0.2x) by moving other numbers to the opposite side of the equation. Here鈥檚 how you can do it:
Start by subtracting 1.86 from both sides of the equation.
  • Original: 0.2x + 1.86 = 5.44
  • Subtract 1.86: 0.2x + 1.86 - 1.86 = 5.44 - 1.86
  • Simplified: 0.2x = 3.58
Remember, whatever you do to one side of the equation, you must do to the other side. This ensures that the equation remains balanced.
Check the Solution
Once you think you have a solution, it鈥檚 crucial to check if it鈥檚 correct. Plug the value back into the original equation to see if both sides are equal:
Replace x with 17.9 into 0.2x + 1.86 = 5.44.
  • Calculation: 0.2(17.9) + 1.86
  • Compute: 3.58 + 1.86 = 5.44
  • Result: both sides equal 5.44
Since both sides match, our solution is correct! Always check your solution to ensure there are no mistakes in your calculations.
Subtraction in Equations
Subtraction is a common step when solving equations. It helps to remove numbers from one side so that the variable stands alone. Here鈥檚 what you need to do:
Identify the number that you need to subtract from both sides.
  • Example: The original equation is 0.2x + 1.86 = 5.44
  • We need to subtract 1.86 from both sides to get rid of it: 0.2x + 1.86 - 1.86 = 5.44 - 1.86
  • Simplify: 0.2x = 3.58
By subtracting the same amount from both sides, the equation stays balanced. This makes sure we're moving closer to isolating the variable.
Division in Equations
After isolating the variable term, you often have to divide to find the actual value of the variable. Here鈥檚 how division helps:
In our example: The equation becomes 0.2x = 3.58 after subtraction.
  • To isolate x, divide both sides by 0.2: 0.2x / 0.2 = 3.58 / 0.2
  • The division simplifies to: x = 17.9
Division is crucial because it scales down the coefficient of the variable (0.2 in this case) to 1, making it simple to read the value of x. Always ensure to divide both sides by the same number to keep the equation balanced.

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Most popular questions from this chapter

In 2009 , Americans generated 243 million tons of municipal solid waste. Food waste was more than \(14 \%\) of this total. Find the minimum amount of food waste generated. Round to the nearest million. (Source: U.S. Statistical Abstract 2012; www.epa.gov)

For exercises 53-62, (a) clear the fractions or decimals and solve. (b) check the direction of the inequality sign. $$ \frac{1}{4} m+\frac{1}{6}>1 $$

For exercises 77-78, \(V=\frac{L W T}{144 \text { in. }^{3}}\) where \(V\) is the volume of lumber in board feet, \(L\) is the length, \(W\) is the width, and \(T\) is the thickness. The units of length, width, and thickness are inches. a. Solve \(V=\frac{L W T}{144 \text { in. }^{3}}\) for \(L\). Include the units of measurement. The thickness of each board in a pile is \(1.5\) in., and the width of each board is \(7.25 \mathrm{in}\). The volume of the boards is 348 board feet. Find the total length of the boards. .. Change this length into feet.

Solve \(V=\frac{L W T}{144}\) for \(W\).

For exercises 9-36, (a) solve. (b) check the direction of the inequality sign. $$ 4 a-12<-60 $$
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