Problem 60 - Introductory Algebra for College Students

Question

Write a linear function, \(f(x)=m x+b,\) satisfying the following conditions: $$f(0)=7 \quad \text { and } \quad f(1)=10$$

Step-by-Step Solution

Answer

The linear function satisfying the given conditions is \(f(x) = 3x + 7.\)

Step 1: Calculate the slope

The slope (m) of the line can be calculated using the formula, \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\). Here, \(x_1=0, y_1=7\) (from f(0)=7) and \(x_2=1, y_2=10\) (from f(1)=10). Substituting these values into the formula, we get: \(m = \frac{{10 - 7}}{{1 - 0}} = 3.\)

Step 2: Find the y-intercept

The y-intercept (b) is the value of 'f' when 'x' is zero. From the condition f(0)=7, it can be seen that the y-intercept 'b' is 7.

Step 3: Writing the linear function

Now substitute 'm' and 'b' into the formula of a linear function. The linear function is, therefore, \(f(x) = mx + b = 3x + 7.\)

Key Concepts

Slope Calculation

The slope of a line, often represented by the letter "m," is a crucial component in understanding linear functions. To calculate the slope, we use the formula \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\). This formula helps us determine the steepness and direction of the line. The \

Y-Intercept

The y-intercept of a linear function is the point where the line crosses the y-axis. It is represented by the letter "b" in the equation\( f(x) = mx + b \). To find the y-intercept, simply substitute the value of \

Linear Equation

A linear equation represents a straight line on a graph and is given in the standard form \( f(x) = mx + b \). This equation contains all the essential information about the line, including its slope and y-intercept. A linear equation can model numerous real-world scenarios, such as predicting trends or understanding relationships between variables in data sets.

• "m" represents the slope of the line, showing how steep it is and in which direction it tilts.
• "b" is the y-intercept, the point where the line intersects the y-axis.
• The variable "x" represents an input or independent variable, influencing the value of "f(x)," the dependent output variable.

Understanding linear equations allows students to recognize patterns and relationships, making these concepts applicable in various aspects, including science and economics. With this fundamental grasp on linear equations, learners can solve more complex algebraic challenges with confidence.