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Chapter 4: Problem 50

Simplify each side of the following equations first, then solve. $$7-16=4 y-3 y+2$$

Short Answer

Expert verified
\(y = -11\)

Step by step solution

01

Simplify Each Side of the Equation

First, let's simplify both sides of the equation. The left side is already simplified: it is simply \(7 - 16\). On the right side, we need to combine the like terms \(4y\) and \(-3y\). This gives us: \[ 4y - 3y = y \]Thus, the equation simplifies to:\[ 7 - 16 = y + 2 \]
02

Simplify the Left Side

Now, simplify the left side of the equation. Calculate \(7 - 16\): \[ 7 - 16 = -9 \]So, the equation becomes:\[ -9 = y + 2 \]
03

Solve for \(y\)

To solve for \(y\), isolate \(y\) on one side of the equation. Subtract 2 from both sides:\[ -9 - 2 = y \] Simplify:\[ y = -11 \]

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Key Concepts

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

Simplifying Expressions
Simplifying expressions is a fundamental skill in solving equations. When you simplify an expression, you aim to make it as straightforward as possible. This often involves combining like terms, removing parentheses by distributing, and carrying out basic arithmetic operations. For example, in the equation from the exercise, both sides of the equation were first simplified. On the left side, there was nothing to simplify further, as it consisted of straightforward arithmetic: \(7 - 16\). The right side required identifying and combining like terms, which leads us to our next key concept.
Like Terms
Like terms are terms that have the same variable raised to the same power. Simplifying an expression by combining like terms reduces the complexity of the equation, making it easier to solve. In our given problem, the terms \(4y\) and \(-3y\) are like terms because both include the variable \(y\). When combined, they simplify to \(y\). Recognizing and grouping like terms is crucial for simplifying equations so we can proceed to solve them effectively.
Isolating Variables
Once the equation is simplified, the next step is isolating the variable. Isolating the variable means getting it alone on one side of the equation. This is typically done by reversing the operations applied to the variable. In the exercise example, the simplified equation was \(-9 = y + 2\). To isolate \(y\), you subtract 2 from both sides, resulting in \(-11 = y\). By eliminating the constant term 2 from the right side, \(y\) becomes isolated, allowing us to solve for its value.
Prealgebra
Prealgebra represents the foundational level of algebra that involves solving basic equations and understanding simple mathematical concepts like addition, subtraction, multiplication, and division. It includes learning skills such as simplifying expressions, identifying like terms, and isolating variables, just like in our exercise here. Prealgebra lays the groundwork for more advanced mathematical concepts, helping students to practice fundamental techniques such as combining like terms and solving for variables in equations. Mastery of these basic concepts is essential for success in higher-level math courses.

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Most popular questions from this chapter

Translate each of the following into an equation, and then solve the equation. The difference of \(x\) and 12 is \(30 .\)

Simplify each expression. $$\left(\frac{13}{21}-\frac{13}{35}\right)^{2}$$

Find three solutions to each of the equations and use them to draw the graph. (GRAPH CANT COPY) $$y=2 x$$

Multiply. $$\frac{1}{4} \cdot 4$$

Match each statement on the left with the property that justifies it on the right. a. Distributive property b. Associative property c. Commutative property d. Commutative and associative properties $$(a+3)+2=a+(3+2)$$
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