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Chapter 3: Problem 61

Perform the indicated operations and simplify as completely as possible. $$4(x-3 y)+2(3 x-y)$$

Short Answer

Expert verified
10x - 14y

Step by step solution

01

Distribute Constants

Distribute the constants 4 and 2 to each term inside the parentheses: 4(x-3y) = 4x - 12y 2(3x - y) = 6x - 2y
02

Combine Like Terms

Combine the like terms from the distributed expressions to simplify: 4x - 12y + 6x - 2y = (4x + 6x) + (-12y - 2y) = 10x - 14y

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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 crucial algebraic concept. It allows you to multiply a single term across terms within parentheses. This property helps simplify expressions and solve equations more efficiently.
Here鈥檚 how it works: Suppose you have an expression like \(a(b+c)\). The distributive property states that you can multiply \(a\) with both \(b\) and \(c\):
  • \(a(b+c) = ab + ac\)
In the original exercise, we used the distributive property to distribute the constants:
  • \(4(x-3y) = 4x - 12y\)
  • \(2(3x-y) = 6x - 2y\)
Applying this property makes the expressions easier to handle and sets the stage for further simplification.
Combining Like Terms
Combining like terms is another vital concept in algebra. Like terms are terms with the same variable raised to the same power. You combine them by adding or subtracting their coefficients.
For example, in the expression \(3x + 5x\), both terms are like terms because they both contain the variable \(x\). So, you can combine them by adding their coefficients:
  • \(3x + 5x = 8x\)
In the exercise, after distributing, we had:
  • \(4x - 12y + 6x - 2y\)
To combine the like terms, we add the coefficients of \(x\) and \(y\) separately:
  • \(4x + 6x = 10x\)
  • \(-12y - 2y = -14y\)
So, the simplified expression is \(10x - 14y\).
Simplifying Expressions
Simplifying expressions means making them as simple as possible. You do this by using algebraic properties like the distributive property and combining like terms. Simplifying makes solving equations and comparing expressions much easier.
In the given exercise, we simplified the expression \(4(x-3y) + 2(3x-y)\) in two steps:
  • First, we used the distributive property to get \(4x - 12y + 6x - 2y\).
  • Then, we combined like terms: \(4x + 6x\) and \(-12y - 2y\), resulting in \(10x - 14y\).
The final simplified expression is \(10x - 14y\). This step-by-step approach helps ensure you correctly apply algebraic rules and avoid common mistakes.

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Most popular questions from this chapter

Solve each of the problems algebraically. That is, set up an equation and solve it. Be sure to clearly label what the variable represents. Round your answer to the nearest tenth where necessary. Metals expand when they are heated. Suppose that the length \(L\) (in centimeters) of a particular metal bar varies with the Celsius temperature \(T\) according to the model $$ L=0.009 T+5.82 $$ (a) Use this model to determine the temperature at which the bar will be \(6.5 \mathrm{cm}\) long. (b) Use this model to determine the length of the bar at a temperature of \(120^{\circ} \mathrm{C}\).

Translate the given phrase or sentence algebraically. \(a\) is at most 5

In a collection of dimes, quarters, and half-dollars, there are 45 coins in all. There are 11 more quarters than half-dollars, and the remaining coins are dimes. If the total value of the coins is \(\$ 11.10,\) how many of each type of coin are there?

Set up an inequality and solve it. Be sure to clearly label what the variable represents. If 9 less than 5 times a number is less than twice the number, how small must the number be?

Solve the inequality and sketch the solution set on a number line. $$-1 \leq 2-3 x<11$$
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