Q27. Simplify [tex]9 m^3 y^2+3 m^2 y[/tex]
Q28. Simplify [tex]\frac{15 m^2+7 m^2-4 m^2}{9 m^2-3 m^2}[/tex]
Q29. Find [tex]x[/tex] in [tex]10-2 x=-18[/tex]
Answer (2)
Factor the first expression: Find the greatest common factor 3 m 2 y and factor it out: 3 m 2 y ( 3 m y + 1 ) .
Simplify the second expression: Combine like terms in the numerator and denominator, then simplify the fraction: 6 m 2 18 m 2 = 3 .
Solve for x in the third expression: Subtract 10 from both sides and divide by -2: x = 14 .
The simplified expressions are 3 m 2 y ( 3 m y + 1 ) , 3 , and x = 14 , respectively. x = 14
Explanation
Overview We will simplify three separate expressions in this problem.
Simplifying the first expression First, let's simplify 9 m 3 y 2 + 3 m 2 y . We need to identify the greatest common factor (GCF) of the two terms. The GCF of 9 and 3 is 3. The GCF of m 3 and m 2 is m 2 . The GCF of y 2 and y is y . Therefore, the GCF of 9 m 3 y 2 and 3 m 2 y is 3 m 2 y . Now, we factor out the GCF from the expression: 9 m 3 y 2 + 3 m 2 y = 3 m 2 y ( 3 m y + 1 ) .
Simplifying the second expression Next, let's simplify 9 m 2 − 3 m 2 15 m 2 + 7 m 2 − 4 m 2 . First, we combine like terms in the numerator: 15 m 2 + 7 m 2 − 4 m 2 = ( 15 + 7 − 4 ) m 2 = 18 m 2 . Then, we combine like terms in the denominator: 9 m 2 − 3 m 2 = ( 9 − 3 ) m 2 = 6 m 2 . So, the expression becomes 6 m 2 18 m 2 . Now, we simplify the fraction by cancelling out the common factor m 2 and reducing the numerical fraction: 6 m 2 18 m 2 = 6 18 = 3 .
Solving for x Finally, let's find x in 10 − 2 x = − 18 . We subtract 10 from both sides of the equation: 10 − 2 x − 10 = − 18 − 10 , which simplifies to − 2 x = − 28 . Then, we divide both sides of the equation by -2: − 2 − 2 x = − 2 − 28 , which simplifies to x = 14 .
Final Answer Therefore, the simplified expressions are:
9 m 3 y 2 + 3 m 2 y = 3 m 2 y ( 3 m y + 1 )
9 m 2 − 3 m 2 15 m 2 + 7 m 2 − 4 m 2 = 3
x = 14
Examples
Factoring and simplifying expressions are fundamental skills in algebra and are used in various real-world applications. For instance, engineers use these skills to simplify complex formulas when designing structures or circuits. Similarly, economists use algebraic simplification to analyze economic models and predict market trends. Solving linear equations is crucial in everyday problem-solving, such as calculating budgets, determining discounts, or planning travel routes.
We simplified three expressions and found their results. The first simplified to 3 m 2 y ( 3 m y + 1 ) , the second simplified to 3 , and we solved for x to find x = 14 .
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