Problem 57 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the general form of the equation of the line that passes through the points. $$(3,2),(0,-1)$$

Step-by-Step Solution

Answer

The general form of the line passing through the points \((3,2)\) and \((0,-1)\) is \(y = x - 1\).

Step 1: Determine the Slope

To find the slope \((m)\) of the line connecting the points \((x_1, y_1)\) and \((x_2, y_2)\), use the equation: \(m = \frac{y_2 - y_1} {x_2 - x_1}\), this gives \(m = \frac{-1-2}{0-3} = 1\).

Step 2: Calculate the Y-Intercept

Knowing the slope \(m\), calculate the Y-intercept \((b)\) using the equation: \(b = y_1 - m * x_1\) with the values \((x_1, y_1)\) as \((3,2)\). This gives: \(b = 2 - 1*3 = -1\).

Step 3: Write the Linear Equation

The line's general equation is expressed in slope-intercept form as \(y = mx + b\). With \(m = 1\) and \(b = -1\), the equation becomes: \(y = x - 1\).

Key Concepts

Slope Calculation

Understanding the slope of a line is crucial as it describes the line's steepness and direction. To calculate the slope, often represented by the symbol m, we use two points that lie on the line. The formula for slope is
\( m = \frac{y_2 - y_1}{x_2 - x_1} \).
For example, if we're given the points
\((3, 2)\) and \((0, -1)\), we subtract the y-coordinate of the first point from the second, and do the same for the x-coordinates:
\( m = \frac{-1 - 2}{0 - 3} \), which simplifies to
\( m = \frac{-3}{-3} = 1 \).
The slope here is 1, indicating that for every unit the line moves horizontally, it also moves up by the same amount vertically. Remember, a negative slope means the line is going downwards as it moves from left to right, while a slope of zero means the line is horizontal.

Y-Intercept

The y-intercept of a line is the point where it crosses the y-axis. This value is signified by the letter b in equations. Once we've calculated the slope, we can find the y-intercept using another point on the line and the formula
\( b = y - m \cdot x \),
where x and y are the coordinates of the known point, and m is the slope. Using the earlier example with the slope of 1 and the point
\((3, 2)\), we insert these into the equation:
\( b = 2 - (1 \cdot 3) = -1 \).
This calculation tells us that the line crosses the y-axis at -1.

• If b is positive, the line crosses the y-axis above the origin.
• If b is negative, it crosses below the origin.
• If b is zero, the line goes through the origin.

Slope-Intercept Form

The slope-intercept form of a linear equation provides a quick way to graph a line and is written as
\( y = mx + b \),
where m is the slope and b is the y-intercept. It's called 'slope-intercept form' because it directly shows both the slope and the y-intercept, which are key to understanding the line's behavior. From our calculations with slope m=1 and y-intercept b=-1, the slope-intercept form of the line is
\( y = x - 1 \).
The beauty of this form is that it's straightforward to plot: start at the y-intercept on the y-axis, and from that point, use the slope to find other points on the line. For instance, since our slope here is 1 (or
\( \frac{1}{1} \)), we move 1 unit up and 1 unit to the right to find another point, then draw the line through these points.