Question 44.1 Simplify:4.1.1 $-x(5x + 1$\]4.1.2 $-2(3x + Y)(5x - 4y$\]4.1.3 $\frac{2x + 1}{4} - \frac{x + 2}{2} - \frac{1}{4}$4.2 Solve For $x$:4.2.1 $3x - 1 = 5$4.2.2 $-6(x - 3) = 2(x + 7) - 5(x - 2$\]

Introduction

Algebraic expressions and equations are fundamental concepts in mathematics, and simplifying them is a crucial step in solving problems. In this article, we will focus on simplifying three different types of algebraic expressions and solving two linear equations. We will use the distributive property, combining like terms, and other algebraic techniques to simplify the expressions and solve the equations.

Simplifying Algebraic Expressions

4.1.1 Simplify: $-x(5x + 1)$

To simplify the expression $-x(5x + 1)$, we need to use the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. In this case, we have $-x(5x + 1)$, which can be rewritten as $-5x^2 - x$. This is because we multiply the $-x$ by each term inside the parentheses, $5x$ and $1$, and then combine the like terms.

4.1.2 Simplify: $-2(3x + y)(5x - 4y)$

To simplify the expression $-2(3x + y)(5x - 4y)$, we need to use the distributive property again. We multiply the $-2$ by each term inside the first parentheses, $3x$ and $y$, and then multiply the result by each term inside the second parentheses, $5x$ and $-4y$. This gives us $-6x^2 - 12xy + 8xy + 8y^2$, which can be rewritten as $-6x^2 - 4xy + 8y^2$.

4.1.3 Simplify: $\frac{2x + 1}{4} - \frac{x + 2}{2} - \frac{1}{4}$

To simplify the expression $\frac{2x + 1}{4} - \frac{x + 2}{2} - \frac{1}{4}$, we need to find a common denominator, which is $4$. We can rewrite each fraction with the common denominator: $\frac{2x + 1}{4} = \frac{2x + 1}{4}$, $\frac{x + 2}{2} = \frac{2x + 4}{4}$, and $\frac{1}{4} = \frac{1}{4}$. Now we can subtract the fractions: $\frac{2x + 1}{4} - \frac{2x + 4}{4} - \frac{1}{4} = \frac{2x + 1 - 2x - 4 - 1}{4} = \frac{-4}{4} = -1$.

Solving Linear Equations

4.2.1 Solve for $x$: $3x - 1 = 5$

To solve the equation $3x - 1 = 5$, we need to isolate the variable $x$. We can do this by adding $1$ to both sides of the equation, which gives us $3x = 6$. Then, we can divide both sides of the equation by $3$, which gives us $x = 2$.

4.2.2 Solve for $x$: $-6(x - 3) = 2(x + 7) - 5(x - 2)$

To solve the equation $-6(x - 3) = 2(x + 7) - 5(x - 2)$, we need to use the distributive property and combine like terms. We can start by multiplying the $-6$ by each term inside the parentheses, $x$ and $-3$, which gives us $-6x + 18$. Then, we can multiply the $2$ by each term inside the parentheses, $x$ and $7$, which gives us $2x + 14$. We can also multiply the $-5$ by each term inside the parentheses, $x$ and $-2$, which gives us $-5x + 10$. Now we can rewrite the equation as $-6x + 18 = 2x + 14 - 5x + 10$. We can combine like terms on the right-hand side of the equation, which gives us $-6x + 18 = -3x + 24$. Now we can add $6x$ to both sides of the equation, which gives us $18 = 24$. This is a contradiction, so there is no solution to the equation.

Conclusion

Q: What is the distributive property, and how is it used in simplifying algebraic expressions?

A: The distributive property is a fundamental concept in algebra that states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. This property is used to simplify algebraic expressions by multiplying the terms inside the parentheses by the term outside the parentheses.

Q: How do I simplify an algebraic expression with multiple terms inside the parentheses?

A: To simplify an algebraic expression with multiple terms inside the parentheses, you need to use the distributive property to multiply each term inside the parentheses by the term outside the parentheses. Then, you can combine like terms to simplify the expression.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2. For example, $2x + 3 = 5$ is a linear equation, while $x^2 + 4x + 4 = 0$ is a quadratic equation.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable by performing the necessary operations to get the variable on one side of the equation and the constant on the other side. You can use addition, subtraction, multiplication, and division to solve the equation.

Q: What is the order of operations, and how is it used in solving algebraic expressions?

A: The order of operations is a set of rules that tells you which operations to perform first when simplifying an algebraic expression. The order of operations is as follows:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify a fraction with a variable in the numerator or denominator?

A: To simplify a fraction with a variable in the numerator or denominator, you need to find the greatest common factor (GCF) of the numerator and denominator and divide both by the GCF.

Q: What is the difference between a rational expression and an irrational expression?

A: A rational expression is an expression that can be written in the form $\frac{p}{q}$, where $p$ and $q$ are polynomials and $q$ is not equal to zero. An irrational expression is an expression that cannot be written in the form $\frac{p}{q}$.

Q: How do I solve a rational equation?

A: To solve a rational equation, you need to find the least common multiple (LCM) of the denominators and multiply both sides of the equation by the LCM. Then, you can simplify the equation and solve for the variable.

Conclusion

In this article, we answered some frequently asked questions on simplifying algebraic expressions and solving linear equations. We covered topics such as the distributive property, simplifying expressions with multiple terms inside the parentheses, linear and quadratic equations, solving linear equations, the order of operations, simplifying fractions with variables, rational and irrational expressions, and solving rational equations. We hope that this article has been helpful in clarifying any doubts you may have had on these topics.