Equation Of A Straight Line Passing Through (4, 3) And (7, 18)

In mathematics, a straight line is defined by its slope and y-intercept. To determine the equation of a straight line, we need to find these two parameters. When given two points on the line, we can use the slope-intercept form (y = mx + c) to derive the equation. This article will guide you through the process of finding the equation of a straight line that passes through the points (4, 3) and (7, 18), expressing the answer in the form y = mx + c, where m and c are integers or fractions in their simplest forms.

The slope-intercept form of a linear equation is expressed as y = mx + c, where:

• y is the dependent variable (typically plotted on the vertical axis).
• x is the independent variable (typically plotted on the horizontal axis).
• m is the slope of the line, representing the rate of change of y with respect to x.
• c is the y-intercept, the point where the line crosses the y-axis (i.e., the value of y when x = 0).

The slope (m) indicates the steepness and direction of the line. A positive slope means the line rises from left to right, while a negative slope means the line falls from left to right. The magnitude of the slope indicates how steep the line is; a larger magnitude means a steeper line. The y-intercept (c) is crucial as it anchors the line's position on the coordinate plane.

To find the equation of a line, we first need to calculate the slope (m) using the coordinates of the two given points. Once we have the slope, we can substitute one of the points into the equation along with the slope to find the y-intercept (c). This process allows us to fully define the line in the slope-intercept form.

The slope of a line passing through two points (x1, y1) and (x2, y2) is calculated using the formula:

m = (y2 - y1) / (x2 - x1)

In our case, the given points are (4, 3) and (7, 18). Let's designate these points as follows:

• (x1, y1) = (4, 3)
• (x2, y2) = (7, 18)

Now, we can substitute these values into the slope formula:

m = (18 - 3) / (7 - 4) m = 15 / 3 m = 5

Thus, the slope of the line is 5. This means that for every one unit increase in x, y increases by 5 units. The positive slope indicates that the line is rising from left to right. This value is a crucial component in defining the equation of the line, as it determines the rate at which the line ascends or descends on the coordinate plane. The slope essentially quantifies the steepness and direction of the line.

Now that we have the slope (m = 5), we can use the slope-intercept form (y = mx + c) and one of the given points to find the y-intercept (c). Let's use the point (4, 3). Substituting the values into the equation:

3 = 5 * 4 + c 3 = 20 + c

To isolate c, subtract 20 from both sides of the equation:

c = 3 - 20 c = -17

Therefore, the y-intercept (c) is -17. This means the line crosses the y-axis at the point (0, -17). The y-intercept is a critical parameter because it anchors the line vertically on the coordinate plane. Knowing this value, along with the slope, allows us to precisely define the line's position and orientation.

Having calculated both the slope (m = 5) and the y-intercept (c = -17), we can now write the equation of the line in slope-intercept form (y = mx + c):

y = 5x - 17

This equation represents the unique straight line that passes through the points (4, 3) and (7, 18). The slope of 5 indicates the line's steepness, and the y-intercept of -17 shows where the line intersects the y-axis. This form is particularly useful because it clearly presents the line's characteristics: how much y changes for each unit change in x, and where the line begins its course on the graph. The equation y = 5x - 17 precisely describes the line's trajectory across the coordinate plane.

To ensure the equation is correct, we can substitute the coordinates of both given points into the equation and check if they satisfy it. First, let's check the point (4, 3):

y = 5x - 17 3 = 5 * 4 - 17 3 = 20 - 17 3 = 3

The equation holds true for the point (4, 3). Now, let's check the point (7, 18):

y = 5x - 17 18 = 5 * 7 - 17 18 = 35 - 17 18 = 18

The equation also holds true for the point (7, 18). Since the equation is satisfied by both points, we can confidently conclude that y = 5x - 17 is the correct equation for the line passing through (4, 3) and (7, 18). This verification step is crucial in ensuring the accuracy of our solution.

In this article, we successfully determined the equation of the straight line that passes through the points (4, 3) and (7, 18). By using the slope formula to calculate the slope (m) and then substituting one of the points into the slope-intercept form (y = mx + c), we found the y-intercept (c). Combining these values, we expressed the equation of the line as y = 5x - 17. This process demonstrates a fundamental method in coordinate geometry for finding the equation of a line given two points. The equation y = 5x - 17 succinctly describes the line's position and direction on the coordinate plane, offering a clear and concise representation of the line's characteristics.

The equation of the straight line that passes through (4, 3) and (7, 18) is y = 5x - 17.