The slope-intercept form of a linear equation is a standard way to represent a straight line. It is expressed as:...
y = mx + b
Where:
- y represents the dependent variable (typically plotted on the vertical axis).
- x represents the independent variable (typically plotted on the horizontal axis).
- m represents the slope of the line, which indicates its steepness and direction.
- b represents the y-intercept, which is the point where the line crosses the y-axis.
Calculating the Slope (m)
The slope of a line is a measure of its steepness. It represents the change in the y-coordinate (rise) divided by the change in the x-coordinate (run) between any two points on the line. We can use the given points (1, 4) and (4, 7) to calculate the slope.
m = (y2 - y1) / (x2 - x1)
m = (7 - 4) / (4 - 1)
m = 3 / 3
m = 1
Determining the y-Intercept (b)
Now that we know the slope (m = 1), we can use one of the given points and the slope-intercept form to find the y-intercept (b). Let's use the point (1, 4).
y = mx + b
4 = 1(1) + b
4 = 1 + b
b = 4 - 1
b = 3
Writing the Equation
Now that we have both the slope (m = 1) and the y-intercept (b = 3), we can plug these values back into the slope-intercept form to obtain the final equation:
y = mx + b
y = 1x + 3
y = x + 3
Therefore, the linear equation in slope-intercept form that contains the points (1, 4) and (4, 7) is y = x + 3.
Visualizing the Line
You can visualize this line by plotting the two points (1, 4) and (4, 7) on a graph and drawing a straight line through them. The line should intersect the y-axis at the point (0, 3), which confirms our calculated y-intercept.
Alternative Methods
While the slope-intercept form is a common way to represent a linear equation, there are other methods available, such as the point-slope form. The point-slope form uses a single point on the line and its slope to determine the equation.
y - y1 = m(x - x1)
You can use either method to arrive at the same linear equation representing the given points.
Key Takeaways
Understanding how to create a linear equation in slope-intercept form is essential in various mathematical and scientific applications. By following the steps outlined above, you can easily determine the equation of a line given two points on it.
Additional Resources
For further exploration of linear equations, consider consulting online resources like Khan Academy, or textbooks on algebra and geometry. You can also utilize graphing calculators or online graphing tools to visualize the lines and verify your results.
Practice Problems
To solidify your understanding, try creating linear equations in slope-intercept form for different sets of points. You can also challenge yourself by finding the equation of a line given its slope and a point on the line.