Problem 91
Question
Problem: Simplify: \(10-6(x-3)\) Incorrect Answer: \(10-6(x-3)\) $$ \begin{aligned} &=4(x-3) \\ &=4 x-12 \end{aligned} $$
Step-by-Step Solution
Verified Answer
The simplified form is \(28 - 6x\).
1Step 1 - Distribute the Negative Sign
Distribute the negative sign through the parentheses. The term inside the parentheses is multiplied by -6, so multiply every term inside by -6: \[10-6(x-3) = 10 - 6x + 18 \]
2Step 2 - Combine Constant Terms
Combine the constants outside the parentheses, 10 and 18: \[10 + 18 - 6x = 28 - 6x \]
Key Concepts
distributive propertycombining like termsconstants in algebra
distributive property
The distributive property is a crucial concept in algebra. It states that when a number is multiplied by a sum or difference inside parentheses, you can distribute the multiplication to each term inside. For example, in the expression \[a(b + c)\], you multiply \(a\) by both \(b\) and \(c\), resulting in \[a \times b + a \times c\]. This helps simplify algebraic expressions by removing parentheses.
In our problem, \(10 - 6(x - 3)\), we need to apply the distributive property to \(-6(x - 3)\). Multiply \(-6\) by both \(x\) and \(-3\). Hence, \(-6 \times x = -6x\) and \(-6 \times -3 = +18\). The expression simplifies to:
\[10 - 6(x - 3) = 10 - 6x + 18\].
In our problem, \(10 - 6(x - 3)\), we need to apply the distributive property to \(-6(x - 3)\). Multiply \(-6\) by both \(x\) and \(-3\). Hence, \(-6 \times x = -6x\) and \(-6 \times -3 = +18\). The expression simplifies to:
\[10 - 6(x - 3) = 10 - 6x + 18\].
combining like terms
Combining like terms is another vital skill in algebra. Like terms are terms that have the same variables raised to the same power. For example, \(3x\) and \(-5x\) are like terms because they both contain \(x\), whereas \(3x\) and \(2y\) are not like terms.
In our problem, after applying the distributive property, we get \[10 - 6x + 18\]. Notice that \(10\) and \(18\) are constants (numbers without variables), and they are like terms. We can combine them by performing addition, \[10 + 18 = 28\]. So, the expression further simplifies to:
\[28 - 6x\].
Always make sure to combine like terms to simplify expressions further.
In our problem, after applying the distributive property, we get \[10 - 6x + 18\]. Notice that \(10\) and \(18\) are constants (numbers without variables), and they are like terms. We can combine them by performing addition, \[10 + 18 = 28\]. So, the expression further simplifies to:
\[28 - 6x\].
Always make sure to combine like terms to simplify expressions further.
constants in algebra
Constants in algebra are numbers on their own, without any attached variables. They remain the same regardless of the variable's value. For instance, in the equation \(2x + 5 = 9\), the numbers \(5\) and \(9\) are constants.
In our problem, after distributing \(-6\) across the terms inside the parentheses, we identify the constants as \(10\) and \(18\). Our goal is to combine these constants to make the expression simpler. Adding these constants gives us \[10 + 18\], which equals \[28\]. This value is independent of the variable \(x\) and thus is a constant in the simplified expression.
Understanding constants is crucial for solving algebra problems efficiently and correctly.
In our problem, after distributing \(-6\) across the terms inside the parentheses, we identify the constants as \(10\) and \(18\). Our goal is to combine these constants to make the expression simpler. Adding these constants gives us \[10 + 18\], which equals \[28\]. This value is independent of the variable \(x\) and thus is a constant in the simplified expression.
Understanding constants is crucial for solving algebra problems efficiently and correctly.
Other exercises in this chapter
Problem 90
\(15+6^{2} \cdot 2 \div 3+9\)
View solution Problem 91
If you use e-mail to communicate with your instructor, your instructor may prefer that you present the information more formally than you would if you were text
View solution Problem 91
\(9^{2} \div 3-12(4-6)\)
View solution Problem 92
Problem: Simplify: \(4-(x+9)\) Incorrect Answer: \(4-(x+9)\) $$ \begin{aligned} &=4-x+9 \\ &=-x+13 \end{aligned} $$
View solution