227. Simplify $9 m^3 y^2 \div 3 m^2 y$
228. Simplify $\frac{15 m^2+7 m^2-4 m^2}{9 m^2-3 m^2}$
229. Find $x$ in $10-2 x=-18$
Answer (2)
Simplify 9 m 3 y 2 ÷ 3 m 2 y to get 3 m y .
Simplify 9 m 2 − 3 m 2 15 m 2 + 7 m 2 − 4 m 2 to get 3 .
Solve 10 − 2 x = − 18 for x to find x = 14 .
The final answers are 3 m y , 3 , and 14 .
Explanation
Introduction We will simplify the given expressions step by step.
Simplifying the first expression Problem 227: Simplify 9 m 3 y 2 ÷ 3 m 2 y . To simplify this expression, we divide the coefficients and subtract the exponents of like variables: 3 m 2 y 9 m 3 y 2 = 3 9 m 3 − 2 y 2 − 1 = 3 m 1 y 1 = 3 m y .
Simplifying the second expression Problem 228: Simplify 9 m 2 − 3 m 2 15 m 2 + 7 m 2 − 4 m 2 . First, combine like terms in the numerator and the denominator: 9 m 2 − 3 m 2 15 m 2 + 7 m 2 − 4 m 2 = ( 9 − 3 ) m 2 ( 15 + 7 − 4 ) m 2 = 6 m 2 18 m 2 . Now, simplify the coefficients and cancel out the common factor m 2 :
6 m 2 18 m 2 = 6 18 = 3.
Solving for x Problem 929: Find x in 10 − 2 x = − 18 . To solve for x , we first isolate the term with x on one side of the equation: 10 − 2 x = − 18 ⇒ − 2 x = − 18 − 10 ⇒ − 2 x = − 28. Now, divide by -2 to solve for x :
x = − 2 − 28 = 14.
Final Answers Therefore, the simplified expressions and the solution for x are: Problem 227: 3 m y Problem 228: 3 Problem 929: x = 14
Examples
These types of algebraic manipulations are useful in various fields. For example, simplifying expressions is crucial in physics when dealing with formulas involving multiple variables. Solving linear equations is fundamental in economics for determining equilibrium points in supply and demand models. Understanding these concepts provides a strong foundation for more advanced mathematical and scientific applications.
The simplified results are 3 m y for the first expression, 3 for the second expression, and x = 14 for the third equation.
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