Find the Equation of a Line Given Two Points in C

Introduction

In this lab, you will learn how to find the equation of a line given two points in C programming. The lab covers three main steps: computing the slope from two points, calculating the y-intercept using the slope and a known point, and finally printing the complete line equation in the form y = mx + b. By the end of this lab, you will have a solid understanding of how to work with lines in analytical geometry using C.

The lab starts by demonstrating how to calculate the slope between two points using the formula (y2 - y1) / (x2 - x1). It then shows how to determine the y-intercept by rearranging the equation y = mx + b to solve for the constant b. Finally, the complete line equation is printed, allowing you to represent the line in the standard form.

Skills Graph

Compute Slope from Two Points

In this step, you will learn how to compute the slope between two points in a C program. The slope represents the steepness of a line and is calculated using the formula: slope = (y2 - y1) / (x2 - x1).

First, create a new C file to implement the slope calculation:

cd ~/project
nano slope_calculation.c

Now, write the following C code to calculate the slope:

#include <stdio.h>

float calculate_slope(float x1, float y1, float x2, float y2) {
    // Check if points are on the same vertical line
    if (x2 - x1 == 0) {
        printf("Undefined slope (vertical line)\n");
        return 0;
    }

    // Calculate slope using the formula: (y2 - y1) / (x2 - x1)
    float slope = (y2 - y1) / (x2 - x1);
    return slope;
}

int main() {
    float x1 = 2.0, y1 = 3.0;  // First point
    float x2 = 5.0, y2 = 7.0;  // Second point

    float slope = calculate_slope(x1, y1, x2, y2);

    printf("Point 1: (%.1f, %.1f)\n", x1, y1);
    printf("Point 2: (%.1f, %.1f)\n", x2, y2);
    printf("Slope: %.2f\n", slope);

    return 0;
}

Compile and run the program:

gcc slope_calculation.c -o slope_calculation
./slope_calculation

Example output:

Point 1: (2.0, 3.0)
Point 2: (5.0, 7.0)
Slope: 1.33

Let's break down the code:

1. The calculate_slope() function takes four parameters: x1, y1, x2, y2
2. It first checks if the points create a vertical line (x2 - x1 = 0)
3. If not a vertical line, it calculates the slope using the standard formula
4. The main() function demonstrates how to use the slope calculation

The slope represents the change in y-coordinate divided by the change in x-coordinate between two points. In this example, the slope is approximately 1.33, which means for every 1 unit change in x, y changes by 1.33 units.

Compute Intercept Using y - mx

In this step, you will learn how to compute the y-intercept of a line using the slope and a known point. The y-intercept is the point where the line crosses the y-axis, and it can be calculated using the equation: b = y - mx.

Continue working in the same project directory and modify the previous C file:

cd ~/project
nano line_equation.c

Write the following C code to calculate the y-intercept:

#include <stdio.h>

float calculate_slope(float x1, float y1, float x2, float y2) {
    if (x2 - x1 == 0) {
        printf("Undefined slope (vertical line)\n");
        return 0;
    }
    return (y2 - y1) / (x2 - x1);
}

float calculate_intercept(float x, float y, float slope) {
    // Calculate y-intercept using the formula: b = y - mx
    float intercept = y - (slope * x);
    return intercept;
}

int main() {
    float x1 = 2.0, y1 = 3.0;  // First point
    float x2 = 5.0, y2 = 7.0;  // Second point

    float slope = calculate_slope(x1, y1, x2, y2);
    float intercept = calculate_intercept(x1, y1, slope);

    printf("Point 1: (%.1f, %.1f)\n", x1, y1);
    printf("Point 2: (%.1f, %.1f)\n", x2, y2);
    printf("Slope: %.2f\n", slope);
    printf("Y-intercept: %.2f\n", intercept);

    return 0;
}

Compile and run the program:

gcc line_equation.c -o line_equation
./line_equation

Example output:

Point 1: (2.0, 3.0)
Point 2: (5.0, 7.0)
Slope: 1.33
Y-intercept: 0.33

Let's break down the new code:

1. The calculate_intercept() function takes three parameters: x, y, and slope
2. It uses the formula b = y - mx to calculate the y-intercept
3. In the main() function, we use the previously calculated slope and a point to find the y-intercept
4. The y-intercept represents where the line crosses the y-axis when x = 0

The calculation shows that for the given points, the line has a slope of 1.33 and a y-intercept of 0.33.

Print Line Equation y = mx + b

In this step, you will learn how to print the complete line equation using the slope and y-intercept calculated in the previous steps. We'll modify the existing C program to display the line equation in the standard form y = mx + b.

Continue working in the same project directory:

cd ~/project
nano line_equation_final.c

Write the following C code to print the line equation:

#include <stdio.h>

float calculate_slope(float x1, float y1, float x2, float y2) {
    if (x2 - x1 == 0) {
        printf("Undefined slope (vertical line)\n");
        return 0;
    }
    return (y2 - y1) / (x2 - x1);
}

float calculate_intercept(float x, float y, float slope) {
    return y - (slope * x);
}

void print_line_equation(float slope, float intercept) {
    printf("Line Equation: y = ");

    // Print slope coefficient
    if (slope == 1) {
        printf("x");
    } else if (slope == -1) {
        printf("-x");
    } else if (slope != 0) {
        printf("%.2fx", slope);
    }

    // Print intercept
    if (intercept > 0) {
        printf(" + %.2f", intercept);
    } else if (intercept < 0) {
        printf(" - %.2f", -intercept);
    }

    printf("\n");
}

int main() {
    float x1 = 2.0, y1 = 3.0;  // First point
    float x2 = 5.0, y2 = 7.0;  // Second point

    float slope = calculate_slope(x1, y1, x2, y2);
    float intercept = calculate_intercept(x1, y1, slope);

    printf("Point 1: (%.1f, %.1f)\n", x1, y1);
    printf("Point 2: (%.1f, %.1f)\n", x2, y2);
    printf("Slope: %.2f\n", slope);
    printf("Y-intercept: %.2f\n", intercept);

    print_line_equation(slope, intercept);

    return 0;
}

Compile and run the program:

gcc line_equation_final.c -o line_equation_final
./line_equation_final

Example output:

Point 1: (2.0, 3.0)
Point 2: (5.0, 7.0)
Slope: 1.33
Y-intercept: 0.33
Line Equation: y = 1.33x + 0.33

Let's break down the new code:

1. The print_line_equation() function handles different cases of slope and intercept
2. It handles special cases like slope of 1, -1, or 0
3. It formats the equation with proper signs for the intercept
4. The function provides a clean, readable representation of the line equation

The code demonstrates how to convert point and slope information into a standard linear equation format.

Summary

In this lab, you first learned how to compute the slope between two points using the formula (y2 - y1) / (x2 - x1). This represents the steepness of the line. Then, you learned how to calculate the y-intercept of the line using the slope and a known point, by rearranging the equation y = mx + b to solve for b. Finally, you combined the slope and intercept to print the complete equation of the line in the form y = mx + b.