Given The Function $f(x) = 3x + 2$, What Is $f(5$\]?A. 17 B. 10 C. 15 D. 21

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving linear equations of the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept. We will use the function $f(x) = 3x + 2$ as a case study to demonstrate the step-by-step process of solving linear equations.

What is a Linear Equation?

A linear equation is an equation in which the highest power of the variable (in this case, $x$) is 1. Linear equations can be written in the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept. The slope represents the rate of change of the function, while the y-intercept represents the point at which the function intersects the y-axis.

The Function $f(x) = 3x + 2$

The function $f(x) = 3x + 2$ is a linear equation with a slope of 3 and a y-intercept of 2. To solve for $f(5)$, we need to substitute $x = 5$ into the function and evaluate the result.

Step 1: Substitute $x = 5$ into the Function

To solve for $f(5)$, we need to substitute $x = 5$ into the function $f(x) = 3x + 2$. This means we replace every instance of $x$ with 5.

$f(5) = 3(5) + 2$

Step 2: Evaluate the Expression

Now that we have substituted $x = 5$ into the function, we need to evaluate the expression. To do this, we multiply 3 by 5 and then add 2.

$f(5) = 15 + 2$

Step 3: Simplify the Expression

The final step is to simplify the expression by combining the terms.

$f(5) = 17$

Conclusion

In this article, we solved the linear equation $f(x) = 3x + 2$ for $f(5)$. We substituted $x = 5$ into the function, evaluated the expression, and simplified the result to get $f(5) = 17$. This demonstrates the step-by-step process of solving linear equations and provides a clear understanding of how to apply this skill in real-world scenarios.

Common Mistakes to Avoid

When solving linear equations, there are several common mistakes to avoid. These include:

• Not substituting the correct value of $x$: Make sure to substitute the correct value of $x$ into the function.
• Not evaluating the expression correctly: Make sure to evaluate the expression correctly by following the order of operations.
• Not simplifying the expression: Make sure to simplify the expression by combining like terms.

Real-World Applications

Linear equations have numerous real-world applications, including:

• Physics: Linear equations are used to describe the motion of objects under constant acceleration.
• Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Linear equations are used to model economic systems and make predictions about future trends.

Practice Problems

To practice solving linear equations, try the following problems:

• Problem 1: Solve the linear equation $f(x) = 2x + 1$ for $f(3)$.
• Problem 2: Solve the linear equation $f(x) = x - 2$ for $f(4)$.
• Problem 3: Solve the linear equation $f(x) = 4x - 3$ for $f(2)$.

Conclusion

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Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (in this case, $x$) is 1. Linear equations can be written in the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to substitute the given value of $x$ into the function and evaluate the expression. Then, simplify the expression by combining like terms.

Q: What is the slope of a linear equation?

A: The slope of a linear equation is the rate of change of the function. It is represented by the coefficient of $x$ in the function.

Q: What is the y-intercept of a linear equation?

A: The y-intercept of a linear equation is the point at which the function intersects the y-axis. It is represented by the constant term in the function.

Q: How do I find the value of a linear equation?

A: To find the value of a linear equation, you need to substitute the given value of $x$ into the function and evaluate the expression. Then, simplify the expression by combining like terms.

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 (in this case, $x$) is 1. A quadratic equation is an equation in which the highest power of the variable (in this case, $x$) is 2.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you need to use the method of substitution or elimination to find the values of the variables.

Q: What is the method of substitution?

A: The method of substitution is a technique used to solve a system of linear equations by substituting the expression for one variable into the other equation.

Q: What is the method of elimination?

A: The method of elimination is a technique used to solve a system of linear equations by adding or subtracting the equations to eliminate one of the variables.

Q: How do I graph a linear equation?

A: To graph a linear equation, you need to plot two points on the graph and draw a line through them.

Q: What is the equation of a line?

A: The equation of a line is a linear equation that represents the relationship between the x and y coordinates of the points on the line.

Q: How do I find the equation of a line?

A: To find the equation of a line, you need to use the slope-intercept form of a linear equation, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Q: What is the slope-intercept form of a linear equation?

A: The slope-intercept form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Q: How do I find the slope of a line?

A: To find the slope of a line, you need to use the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.

Q: What is the y-intercept of a line?

A: The y-intercept of a line is the point at which the line intersects the y-axis. It is represented by the constant term in the equation of the line.

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By understanding the concepts of linear equations, including the slope and y-intercept, students can develop a clear understanding of how to solve linear equations and apply this skill in real-world scenarios. Remember to practice solving linear equations and to use the methods of substitution and elimination to solve systems of linear equations. With practice and patience, students can become proficient in solving linear equations and unlock a world of possibilities in mathematics and beyond.