Problem 45 - Algebra 1

Question

Write an equation of the line that passes through the points. (3,7),(7,3)

Step-by-Step Solution

Answer

The equation of the line passing through the points (3,7) and (7,3) is \(y = -x + 10\).

Step 1: Find the slope

First, find the slope using the formula \(m = \frac{{y2 - y1}}{{x2 - x1}}\), where (x1, y1) and (x2, y2) are the given points. Here, (x1, y1) = (3,7) and (x2, y2) = (7,3), so \(m = \frac{{3 - 7}}{{7 - 3}} = -1\).

Step 2: Finding the y-intercept

To find the y-intercept, use the equation of a line in slope-intercept form, which is y = mx + b. Substitute any point\'s coordinates (x, y) and the slope m to find b. In this case, using point (3, 7) and m = -1, we obtain \(b = 7 - (-1) * 3 = 7 + 3 = 10\).

Step 3: Writing the Final Equation

Substitute the values of m and b into the equation y = mx + b. The final line's equation that passes through the points (3,7) and (7,3) would then be written as y = -1x + 10, which simplifies to y = -x + 10.

Key Concepts

Understanding the Slope Formula

Finding the slope of a line is crucial for determining the equation of the line. The slope tells us how steep the line is and in which direction it extends. The slope formula is given by:

• \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\)

In this formula, \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on the line. The numerator \((y_2 - y_1)\) represents the change in the y-values, while the denominator \((x_2 - x_1)\) represents the change in the x-values. Essentially, the slope measures the ratio of the vertical change to the horizontal change between two points.

In our example, the points (3,7) and (7,3) were used to find the slope. By plugging these values into the slope formula, we calculated:

• \(m = \frac{{3 - 7}}{{7 - 3}} = -1\)

This negative slope of -1 indicates that the line falls downwards as we move from left to right.

Determining the Y-Intercept

The y-intercept is the point where the line crosses the y-axis, which gives us a valuable insight into the position of the line on a graph. To find it, we use the slope-intercept form of a linear equation:

• \(y = mx + b\)

Here, \(b\) is the y-intercept, and \(m\) is the slope, which we already know. By substituting a point on the line and the slope into the equation, we can solve for \(b\). For example, using the point (3,7):

• Plug \(7\) in for \(y\), \(-1\) for \(m\) (the slope), and \(3\) for \(x\):
• \(7 = -1 \times 3 + b\)
• Solving for \(b\) gives \(b = 10\)

This means the line intersects the y-axis at the point (0,10). The y-intercept allows us to understand the starting point of the line on a graph.

The Slope-Intercept Form

The slope-intercept form is an efficient and widely-used form for describing a straight line. It is represented as:

• \(y = mx + b\)

This equation directly reveals two vital properties of the line: the slope \(m\), which indicates the incline or decline of the line, and the y-intercept \(b\), which tells us where the line crosses the y-axis. The slope-intercept form is incredibly useful in quickly sketching the graph of the line because it immediately shows both the direction and the starting position of the line.

In our example, we found the slope \(m = -1\) and the y-intercept \(b = 10\). Plugging these values into the slope-intercept form, we obtained the equation:

• \(y = -x + 10\)

This equation can easily be graphed, revealing a line that descends from left to right, crossing the y-axis at 10.