What Would Be A Good First Step For Solving The Equation Given Below?$\frac{3}{x+9}=8$A. Multiply Both Sides By 9 B. Subtract 8 From Both Sides C. Multiply Both Sides By $x+9$ D. Add $x+9$ To Both Sides

What would be a good first step for solving the equation given below?

Understanding the Equation

The given equation is $\frac{3}{x+9}=8$. To solve this equation, we need to isolate the variable $x$. The first step in solving an equation is to simplify it and make it easier to work with.

Simplifying the Equation

Looking at the equation, we can see that the denominator is $x+9$. To simplify the equation, we can multiply both sides by the denominator, which is $x+9$. This will eliminate the fraction and make it easier to work with.

Choosing the Correct Option

Now, let's look at the options given:

A. Multiply both sides by 9 B. Subtract 8 from both sides C. Multiply both sides by $x+9$ D. Add $x+9$ to both sides

The correct option is C. Multiply both sides by $x+9$. This is because multiplying both sides by the denominator will eliminate the fraction and make it easier to work with.

Why is this the correct option?

Multiplying both sides by $x+9$ is the correct option because it will eliminate the fraction and make it easier to work with. This is a common technique used in algebra to simplify equations and make them easier to solve.

What happens if we choose the wrong option?

If we choose the wrong option, we may end up with a more complicated equation or an equation that is not true. For example, if we choose option A, we would be multiplying both sides by 9, which is not the denominator of the fraction. This would not eliminate the fraction and would make the equation more complicated.

Conclusion

In conclusion, the first step in solving the equation $\frac{3}{x+9}=8$ is to multiply both sides by the denominator, which is $x+9$. This will eliminate the fraction and make it easier to work with.

Step-by-Step Solution

Here is the step-by-step solution to the equation:

1. Multiply both sides by $x+9$: $\frac{3}{x+9} \cdot (x+9) = 8 \cdot (x+9)$
2. Simplify the equation: $3 = 8(x+9)$
3. Distribute the 8: $3 = 8x + 72$
4. Subtract 72 from both sides: $-69 = 8x$
5. Divide both sides by 8: $x = -\frac{69}{8}$

Final Answer

The final answer is $x = -\frac{69}{8}$.

Why is this the correct answer?

This is the correct answer because it satisfies the original equation. We can plug in $x = -\frac{69}{8}$ into the original equation and verify that it is true.

Conclusion

In conclusion, the first step in solving the equation $\frac{3}{x+9}=8$ is to multiply both sides by the denominator, which is $x+9$. This will eliminate the fraction and make it easier to work with. The final answer is $x = -\frac{69}{8}$.
Frequently Asked Questions (FAQs) about Solving Equations

Q: What is the first step in solving an equation?

A: The first step in solving an equation is to simplify it and make it easier to work with. This may involve multiplying both sides by the denominator, adding or subtracting the same value to both sides, or other techniques.

Q: Why do we need to simplify the equation?

A: Simplifying the equation makes it easier to work with and helps us to isolate the variable. By simplifying the equation, we can make it easier to solve and find the value of the variable.

Q: What is the difference between multiplying and dividing both sides of an equation?

A: Multiplying both sides of an equation is the same as multiplying both sides by 1. This does not change the value of the equation. Dividing both sides of an equation is the same as multiplying both sides by the reciprocal of the divisor. This can change the value of the equation.

Q: Can we add or subtract the same value to both sides of an equation?

A: Yes, we can add or subtract the same value to both sides of an equation. This is a common technique used in algebra to simplify equations and make them easier to solve.

Q: What is the order of operations when solving an equation?

A: The order of operations when solving an equation is:

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 we know if an equation is true or false?

A: We can determine if an equation is true or false by plugging in a value for the variable and evaluating the equation. If the equation is true, the value we plugged in will make the equation true. If the equation is false, the value we plugged in will make the equation false.

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. A quadratic equation is an equation in which the highest power of the variable is 2.

Q: Can we solve a quadratic equation by factoring?

A: Yes, we can solve a quadratic equation by factoring. Factoring involves expressing the quadratic equation as a product of two binomials.

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations. It is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: How do we choose between the quadratic formula and factoring?

A: We choose between the quadratic formula and factoring based on the complexity of the equation. If the equation is simple and can be easily factored, we use factoring. If the equation is complex and cannot be easily factored, we use the quadratic formula.

Q: What is the difference between a system of linear equations and a system of quadratic equations?

A: A system of linear equations is a set of linear equations that are solved simultaneously. A system of quadratic equations is a set of quadratic equations that are solved simultaneously.

Q: Can we solve a system of linear equations by substitution?

A: Yes, we can solve a system of linear equations by substitution. Substitution involves solving one equation for one variable and then substituting that value into the other equation.

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

A: A linear inequality is an inequality in which the highest power of the variable is 1. A quadratic inequality is an inequality in which the highest power of the variable is 2.

Q: Can we solve a quadratic inequality by factoring?

A: Yes, we can solve a quadratic inequality by factoring. Factoring involves expressing the quadratic inequality as a product of two binomials.

Q: What is the difference between a rational inequality and a quadratic inequality?

A: A rational inequality is an inequality that involves a rational expression. A quadratic inequality is an inequality that involves a quadratic expression.

Q: Can we solve a rational inequality by factoring?

A: Yes, we can solve a rational inequality by factoring. Factoring involves expressing the rational inequality as a product of two binomials.

Q: What is the difference between a polynomial inequality and a rational inequality?

A: A polynomial inequality is an inequality that involves a polynomial expression. A rational inequality is an inequality that involves a rational expression.

Q: Can we solve a polynomial inequality by factoring?

A: Yes, we can solve a polynomial inequality by factoring. Factoring involves expressing the polynomial inequality as a product of two binomials.

Q: What is the difference between a system of polynomial inequalities and a system of rational inequalities?

A: A system of polynomial inequalities is a set of polynomial inequalities that are solved simultaneously. A system of rational inequalities is a set of rational inequalities that are solved simultaneously.

Q: Can we solve a system of polynomial inequalities by substitution?

A: Yes, we can solve a system of polynomial inequalities by substitution. Substitution involves solving one inequality for one variable and then substituting that value into the other inequality.

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

A: A linear equation is an equation in which the highest power of the variable is 1. A nonlinear equation is an equation in which the highest power of the variable is greater than 1.

Q: Can we solve a nonlinear equation by factoring?

A: No, we cannot solve a nonlinear equation by factoring. Nonlinear equations require more advanced techniques, such as the quadratic formula or numerical methods.

Q: What is the difference between a linear inequality and a nonlinear inequality?

A: A linear inequality is an inequality in which the highest power of the variable is 1. A nonlinear inequality is an inequality in which the highest power of the variable is greater than 1.

Q: Can we solve a nonlinear inequality by factoring?

A: No, we cannot solve a nonlinear inequality by factoring. Nonlinear inequalities require more advanced techniques, such as the quadratic formula or numerical methods.

Q: What is the difference between a system of linear inequalities and a system of nonlinear inequalities?

A: A system of linear inequalities is a set of linear inequalities that are solved simultaneously. A system of nonlinear inequalities is a set of nonlinear inequalities that are solved simultaneously.

Q: Can we solve a system of nonlinear inequalities by substitution?

A: No, we cannot solve a system of nonlinear inequalities by substitution. Nonlinear inequalities require more advanced techniques, such as the quadratic formula or numerical methods.

Q: What is the difference between a linear equation and a nonlinear equation in three variables?

A: A linear equation in three variables is an equation in which the highest power of each variable is 1. A nonlinear equation in three variables is an equation in which the highest power of at least one variable is greater than 1.

Q: Can we solve a nonlinear equation in three variables by factoring?

A: No, we cannot solve a nonlinear equation in three variables by factoring. Nonlinear equations in three variables require more advanced techniques, such as the quadratic formula or numerical methods.

Q: What is the difference between a linear inequality and a nonlinear inequality in three variables?

A: A linear inequality in three variables is an inequality in which the highest power of each variable is 1. A nonlinear inequality in three variables is an inequality in which the highest power of at least one variable is greater than 1.

Q: Can we solve a nonlinear inequality in three variables by factoring?

A: No, we cannot solve a nonlinear inequality in three variables by factoring. Nonlinear inequalities in three variables require more advanced techniques, such as the quadratic formula or numerical methods.

Q: What is the difference between a system of linear inequalities and a system of nonlinear inequalities in three variables?

A: A system of linear inequalities in three variables is a set of linear inequalities that are solved simultaneously. A system of nonlinear inequalities in three variables is a set of nonlinear inequalities that are solved simultaneously.

Q: Can we solve a system of nonlinear inequalities in three variables by substitution?

A: No, we cannot solve a system of nonlinear inequalities in three variables by substitution. Nonlinear inequalities in three variables require more advanced techniques, such as the quadratic formula or numerical methods.

Conclusion

In conclusion, solving equations and inequalities requires a variety of techniques, including factoring, the quadratic formula, and numerical methods. The choice of technique depends on the complexity of the equation or inequality and the number of variables involved.