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Chapter 2: Problem 18

For exercises 11-46, (a) solve. (b) check. $$ 0.29=y+0.81 $$

Short Answer

Expert verified
y = -0.52

Step by step solution

01

- Isolate the variable

First, subtract 0.81 from both sides of the equation to isolate the variable y.\[0.29 - 0.81 = y + 0.81 - 0.81\]This simplifies to:\[0.29 - 0.81 = y\]
02

- Perform the subtraction

Now, calculate the subtraction on the left-hand side of the equation.\[0.29 - 0.81 = -0.52\]So,\[y = -0.52\]
03

- Check the solution

To verify the solution, substitute \(y = -0.52\) back into the original equation.\[0.29 = -0.52 + 0.81\]Calculate the right-hand side:\[0.29 = 0.29\]Since both sides of the equation are equal, our solution is correct.

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Key Concepts

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

Isolating Variables
When solving linear equations, the goal is to isolate the variable. This usually means getting the variable by itself on one side of the equation. Consider an equation like the one in the exercise: $$0.29 = y + 0.81$$ First, we need to isolate y. To do this, we need to 'move' everything else to the other side of the equation. This is done through various algebraic operations.

For our equation, we subtract 0.81 from both sides. Why subtract? Because we want to cancel out the +0.81 on the right side. The subtraction leaves us with:

$$0.29 - 0.81 = y$$
This step is essential in solving the equation correctly.
  • Always perform the same operation on both sides of the equation
  • This keeps the equation balanced
  • Follow the correct order of operations
With the variable isolated, we can move on to the next step.
Subtraction in Algebra
Subtraction in algebra works under the same principles as regular subtraction. When we subtract numbers, we are essentially decreasing the value of one number by the value of another.

Consider our equation after isolating y: $$0.29 - 0.81 = y$$ Performing the subtraction is straightforward:
  • Calculate the difference between 0.29 and 0.81
  • In this case, it becomes evident that the result is a negative value since 0.81 is larger than 0.29
So, $$0.29 - 0.81 = -0.52$$ Thus, we find that $$y = -0.52$$ . It's important to understand that performing the correct subtraction operation is crucial to finding the right value of the isolated variable.
Checking Solutions
After finding the value of the variable, it's a good practice to check the solution. This step ensures our calculations are correct.

We substitute the value back into the original equation to see if it holds true. Our solution is $$ y = -0.52$$ Let's substitute $$ y = -0.52$$ into the original equation: $$0.29 = -0.52 + 0.81$$ Now, perform the addition on the right-hand side: $$-0.52 + 0.81 = 0.29$$ As we can see, both sides of the equation are equal: $$0.29 = 0.29$$ .
  • This confirms that our solution is correct
  • It's always a reliable step to re-verify your solution
Checking solutions is crucial for ensuring accuracy in math problem-solving.

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Most popular questions from this chapter

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

Each of the four sides of a square is the same length. If \(P\) is the perimeter of the square and \(S\) is the length of a side, write a formula in \(P\) and \(S\) for the perimeter of a square and solve this formula for \(S\).

For exercises 83-84, a rectangular solid has \(C\) corners, \(E\) edges, and \(F\) faces. a. Solve \(C-E+F=2\) for \(F\). b. A rectangular book has eight corners \((C)\) and 12 edges \((E)\). Find the number of faces \((F)\).

For problems \(89-92\), do the arithmetic with a calculator. The volume \(V\) of an atrium in a hotel is \(648,000 \mathrm{ft}^{3}\). The rectangular floor of the atrium is \(90 \mathrm{ft}\) wide and \(120 \mathrm{ft}\) long. Solve the formula \(V=L W H\) for \(H\), and use it to find the height of the atrium.

For exercises 53-62, (a) clear the fractions or decimals and solve. (b) check the direction of the inequality sign. $$ \frac{3}{4}(2 u+8)<\frac{5}{6}(3 u-12) $$
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