Simplifying Expressions And Solving Equations A Comprehensive Guide

This comprehensive guide delves into the fundamental concepts of simplifying algebraic expressions and solving equations, providing step-by-step solutions and clear explanations. We'll explore techniques for combining like terms, factoring, and isolating variables, empowering you to tackle a wide range of mathematical problems. Whether you're a student learning algebra for the first time or a seasoned mathematician looking for a refresher, this guide offers valuable insights and practical strategies to enhance your understanding and skills.

Q27. Simplify $9m^3y^2 + 3m^2y$

In this section, we will address the simplification of the algebraic expression $9m^3y^2 + 3m^2y$. Simplifying algebraic expressions is a fundamental skill in mathematics, allowing us to rewrite expressions in a more concise and manageable form. This often involves identifying common factors and applying the distributive property in reverse, a process known as factoring. In this particular case, our focus is on identifying the greatest common factor (GCF) of the terms $9m^3y^2$ and $3m^2y$. The GCF is the largest factor that divides evenly into both terms. To find the GCF, we consider the coefficients and the variables separately.

Identifying the Greatest Common Factor (GCF)

First, let's look at the coefficients, which are 9 and 3. The GCF of 9 and 3 is 3 because 3 is the largest number that divides both 9 and 3 without leaving a remainder. Next, we examine the variables. We have $m^3$ in the first term and $m^2$ in the second term. The GCF of $m^3$ and $m^2$ is $m^2$, as $m^2$ is the highest power of $m$ that is a factor of both terms. Similarly, we have $y^2$ in the first term and $y$ in the second term. The GCF of $y^2$ and $y$ is $y$, as $y$ is the highest power of $y$ that is a factor of both terms. Combining these, the GCF of the entire expression is $3m^2y$.

Factoring out the GCF

Now that we have identified the GCF, we can factor it out of the original expression. Factoring involves dividing each term by the GCF and writing the expression as a product of the GCF and the resulting quotient. So, we divide $9m^3y^2$ by $3m^2y$, which gives us $3my$. We also divide $3m^2y$ by $3m^2y$, which gives us 1. Therefore, we can rewrite the expression as $3m^2y(3my + 1)$. This is the simplified form of the original expression, as we have factored out the greatest common factor and cannot simplify it further. This process of identifying and factoring out the GCF is a crucial technique in simplifying algebraic expressions and is used extensively in various mathematical contexts, including solving equations and working with polynomials. Understanding this concept thoroughly allows for more efficient and accurate manipulation of algebraic expressions.

Solution:

$9m^3y^2 + 3m^2y = 3m^2y(3my + 1)$

Q28. Simplify $\frac{15m^2 + 7m^2 - 4m^2}{9m^2 - 3m^2}$

This section focuses on simplifying the algebraic fraction $\frac{15m^2 + 7m^2 - 4m^2}{9m^2 - 3m^2}$. Simplifying algebraic fractions involves reducing the fraction to its simplest form by combining like terms in both the numerator and the denominator, and then canceling out any common factors. The process is similar to simplifying numerical fractions, but with the added complexity of dealing with variables and algebraic expressions. To simplify the given expression, we first need to simplify the numerator and the denominator separately by combining like terms. Like terms are terms that have the same variable raised to the same power; in this case, all the terms involve $m^2$, so they are like terms and can be combined through addition and subtraction.

Simplifying the Numerator and Denominator

Let's start by simplifying the numerator, which is $15m^2 + 7m^2 - 4m^2$. We combine the coefficients of the $m^2$ terms: $15 + 7 - 4 = 18$. So, the simplified numerator is $18m^2$. Next, we simplify the denominator, which is $9m^2 - 3m^2$. Again, we combine the coefficients of the $m^2$ terms: $9 - 3 = 6$. So, the simplified denominator is $6m^2$. Now, the original fraction is reduced to $\frac{18m^2}{6m^2}$. This simplified fraction is much easier to work with and allows us to see the common factors more clearly.

Canceling Common Factors

Once we have simplified the numerator and denominator, the next step is to cancel out any common factors. In this case, both the numerator and the denominator have $m^2$ as a common factor, which can be canceled out. Additionally, the coefficients 18 and 6 have a common factor of 6. Dividing both the numerator and the denominator by $m^2$, we get $\frac{18}{6}$. Now, we simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 6. Dividing 18 by 6 gives us 3, and dividing 6 by 6 gives us 1. Therefore, the simplified fraction is $\frac{3}{1}$, which is simply 3. This final simplification demonstrates the power of combining like terms and canceling common factors to arrive at the most concise form of the algebraic fraction. Understanding these techniques is essential for handling more complex algebraic expressions and equations, and it forms a cornerstone of algebraic manipulation.

Solution:

$\frac{15m^2 + 7m^2 - 4m^2}{9m^2 - 3m^2} = \frac{18m^2}{6m^2} = 3$

Q29. Find $x$ in $10 - 2x = -18$

In this section, we aim to solve the linear equation $10 - 2x = -18$ for the variable $x$. Solving linear equations is a fundamental skill in algebra, and it involves isolating the variable on one side of the equation to determine its value. The basic strategy is to perform the same operations on both sides of the equation to maintain balance while moving terms around until the variable is isolated. This often involves using inverse operations to undo the operations applied to the variable. In this particular equation, we need to isolate $x$ by undoing the subtraction and multiplication operations.

Isolating the Variable

To begin, we want to isolate the term containing $x$, which is $-2x$. To do this, we need to eliminate the 10 on the left side of the equation. We can do this by subtracting 10 from both sides of the equation. This maintains the balance of the equation, as we are performing the same operation on both sides. Subtracting 10 from both sides of $10 - 2x = -18$ gives us $10 - 2x - 10 = -18 - 10$, which simplifies to $-2x = -28$. Now, we have a simpler equation where the term with $x$ is isolated on the left side.

Solving for x

The next step is to solve for $x$ by undoing the multiplication. Currently, $x$ is being multiplied by -2. To isolate $x$, we need to divide both sides of the equation by -2. Dividing both sides of $-2x = -28$ by -2 gives us $\frac{-2x}{-2} = \frac{-28}{-2}$, which simplifies to $x = 14$. Thus, we have found the value of $x$ that satisfies the original equation. To ensure our solution is correct, we can substitute $x = 14$ back into the original equation and verify that it holds true. Substituting $x = 14$ into $10 - 2x = -18$ gives us $10 - 2(14) = -18$, which simplifies to $10 - 28 = -18$, and further simplifies to $-18 = -18$. Since this is a true statement, our solution of $x = 14$ is correct. This process of solving linear equations by isolating the variable is a critical skill in algebra and is essential for tackling more complex mathematical problems. Understanding and mastering these techniques provides a strong foundation for further mathematical studies.

Solution:

$10 - 2x = -18$

$-2x = -28$

$x = 14$