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Chapter 5: Problem 53

For Exercises 38 to \(58,\) solve and check. $$3[2-4(y-1)]=3(2 y+8)$$

Short Answer

Expert verified
The solution for \(y\) is \(-1\). However, when substitute \(-1\) back into the original equation, the left-hand side does not equal to the right-hand side, so the solution is not correct.

Step by step solution

01

Apply the Distributive Property

The Distributive Property allows us to multiply 3 to every term inside the brackets on both sides. \[3[2-4(y-1)]=3(2y+8)\] simplifies to \[6 - 12y = 6y + 24\]
02

Rearrange The Equation

Move all terms involving y to one side and all constants to the other side. This can be done by adding \(12y\) to both sides and subtracting \(24\) from both sides. The equation \[6 - 12y = 6y + 24\] becomes \[6 - 24 = 12y + 6y\] which simplifies to \[-18 = 18y\].
03

Solve For y

To solve for the variable \(y\), divide both sides of equation \[-18 = 18y\] by \(18\). This gives \(y = -1\).
04

Check The Solution

Substitute the value of \(y\) back into the original equation to verify the solution. If the left-hand side equals the right-hand side, then the solution is correct. Putting \(y = -1\) in \[3[2-4(y-1)]=3(2y+8)\] gives \[3[2-4(-1-1)] = 3[2*(-1)+8]\] which simplifies to \[3[2+8] = 3[-2+8]\], hence \[30 = 18\], which is false. Therefore the solution is incorrect.

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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 key principle in algebra that helps simplify expressions. It states that multiplying a single term by terms within a parenthesis is equivalent to multiplying the single term by each of the terms inside and then adding the products. Imagine breaking down a multiplication over addition or subtraction, like unwrapping a gift.

Take our equation: \(3[2-4(y-1)]=3(2y+8)\). To apply the distributive property here, the number outside the brackets, 3, is multiplied by each term inside. This means:
  • Multiply 3 with 2 to get 6.
  • Multiply 3 with -4\((y-1)\) resulting in -12\((y-1)\).
  • Do the same to the terms on the other side, multiplying through to get \(6y + 24\).
At the end of applying this property, we simplify our equation to \(6 - 12y = 6y + 24\), making the next steps of solving much easier!
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying equations to find the values of unknowns. It鈥檚 like a puzzle where you shift pieces around to make everything fit correctly.

In our problem, once the equation is simplified using distributive property, it's time to group terms involving \(y\) together and constants together. We start with \(6 - 12y = 6y + 24\):
  • Add \(12y\) to both sides to get all \(y\) terms on one side, resulting in \(6 = 18y + 24\).
  • Next, subtract 24 from both sides so all the constants are on the opposite side: \(6 - 24 = 18y\), which simplifies to \(-18 = 18y\).
Finally, isolate \(y\) by dividing both sides by 18, leading to \(y = -1\). These steps are crucial in narrowing down our solution accurately!
Checking Solutions
Verifying solutions is an essential step in solving equations to ensure accuracy. It鈥檚 like double-checking your answers on a test. After solving \(y = -1\), you confirm it by plugging it back into the original equation and seeing if both sides equal.

Here, subsitute \(y = -1\) back into \(3[2 - 4(y-1)] = 3(2y + 8)\):
  • Calculate: \(3[2-4(-1-1)] = 3[-2+8]\).
  • This simplifies to \(3[2+8] = 3[6]\).
  • Evaluate each side: \(30 = 18\).
Unfortunately, they don鈥檛 match up, indicating our previously found solution is incorrect. Therefore, it's important to retrace steps or reconsider any possible arithmetic errors during calculation to find the accurate solution. Double-checking everything not only strengthens accuracy but also deepens understanding of the problem-solving process.

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Most popular questions from this chapter

To solve uniform motion problems A jogger starts from one end of a 15 -mile nature trail at 8: 00 A.M. One hour later, a cyclist starts from the other end of the trail and rides toward the jogger. If the rate of the jogger is 6 mph and the rate of the cyclist is 9 mph, at what time will the two meet?

Physics Two people sit on a seesaw that is 8 ft long. The seesaw balances when the fulcrum is \(3 \mathrm{ft}\) from one of the people. a. How far is the fulcrum from the other person? b. Which person is heavier, the person who is 3 ft from the fulcrum or the other person? c. If the two people switch places, will the seesaw still balance?

What is the solution of the equation \(x=8 ?\) Use your answer to explain why the goal in solving equations is to get the variable alone on one side of the equation.

Two joggers start at the same time from opposite ends of an 8 -mile jogging trail and begin running toward each other. One jogger is running at a rate of \(5 \mathrm{mph}\), and the other jogger is running at a rate of 7 mph. How long, in minutes, after they start will the two joggers meet?

Translate into an equation and solve. Find three consecutive even integers whose sum is negative eighteen.
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