factor out the coefficient of the variable 1/4y +3/8
The Correct Answer and Explanation is:
Step-by-step solution:
- Start with the expression: 14y+38\frac{1}{4}y + \frac{3}{8}41y+83
- Factor out 14\frac{1}{4}41 from the expression. This means dividing each term by 14\frac{1}{4}41 and placing the result inside parentheses: 14(y+38÷14)\frac{1}{4}\left(y + \frac{3}{8} \div \frac{1}{4}\right)41(y+83÷41)
- Simplify the division: 38÷14=38×41=128=32\frac{3}{8} \div \frac{1}{4} = \frac{3}{8} \times \frac{4}{1} = \frac{12}{8} = \frac{3}{2}83÷41=83×14=812=23
- Substitute back into the expression: 14(y+32)\frac{1}{4}(y + \frac{3}{2})41(y+23)
Final Answer:
14(y+32)\frac{1}{4}(y + \frac{3}{2})41(y+23)
Textbook-style Explanation
Factoring is a fundamental algebraic technique used to simplify expressions and solve equations. In the expression 14y+38\frac{1}{4}y + \frac{3}{8}41y+83, the goal is to factor out the coefficient of the variable term, which is 14\frac{1}{4}41. This technique involves rewriting the original expression as a product of a common factor and a simplified expression inside parentheses.
Begin by identifying the coefficient of the variable term yyy, which is 14\frac{1}{4}41. Factoring this out means expressing both terms of the original expression in terms of 14\frac{1}{4}41. To do so, divide each term by 14\frac{1}{4}41, keeping in mind that dividing fractions requires multiplying by the reciprocal.
The second term, 38\frac{3}{8}83, divided by 14\frac{1}{4}41 becomes 38÷14=38×41=128=32\frac{3}{8} \div \frac{1}{4} = \frac{3}{8} \times \frac{4}{1} = \frac{12}{8} = \frac{3}{2}83÷41=83×14=812=23. Now, rewrite the original expression by factoring 14\frac{1}{4}41 from both terms:14(y+32)\frac{1}{4}(y + \frac{3}{2})41(y+23)
This result represents the same expression in a factored form. Factoring helps reveal the structure of expressions, making it easier to work with them in more complex algebraic problems, especially when solving equations or simplifying expressions. It is also useful when solving word problems, graphing linear functions, or preparing for higher-level mathematics such as calculus, where factoring is frequently used for simplification.
In this case, factoring out 14\frac{1}{4}41 simplifies the expression into a single term multiplied by a binomial. This form is often preferred when solving equations or analyzing algebraic relationships.
