Problem 49 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of line that passes through the given points. \((5,2)\) and \((8,2)\)

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form that passes through the points \((5,2)\) and \((8,2)\) is \(y = 2\).

Step 1: Calculate the Slope

The slope formula is \((y_2 - y_1)/(x_2 - x_1)\). Substitute the given points into the formula: \((2 - 2)/(8 - 5) = 0/3 = 0\). So, the slope (m) of the line is 0.

Step 2: Use the Point-Slope Form to Find the Y-Intercept

The general form of the point-slope equation is \(y - y_1 = m(x - x_1)\). Choose one of the given points, say \((5,2)\), and replace the values of x, y, and m into the equation: \(2 - 2 = 0(5 - 5)\). Solve for y to find the y-intercept (b): 2 = 0. So, the y-intercept is 2.

Step 3: Write the Equation in Slope-Intercept Form

The slope-intercept form of the line is \(y = mx + b\). Replace m by 0 and b by 2, to get: \(y = 0x + 2\), this simplifies to \(y = 2\).

Key Concepts

Equation of a Line

An equation of a line is a mathematical expression that describes all the points that lie on that line. In algebra, the most common form used for this is the "slope-intercept form," denoted as \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept. This form is particularly useful because it directly shows the rate at which \(y\) changes with \(x\) (slope) as well as the point where the line crosses the y-axis (y-intercept).
It's straightforward for graphing and understanding how changes in \(x\) affect \(y\). To write the equation of a line using this form, you'll need the slope and at least one point that lies on the line. From there, you have everything you need to capture the essence of the line in a simple equation.

Slope

The slope of a line measures its steepness and direction. It's calculated using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), where \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on the line.

• A positive slope means the line is rising as it moves from left to right.
• A negative slope indicates the line is falling as it moves from left to right.
• A zero slope suggests the line is perfectly horizontal.
• A slope that is undefined signifies a vertical line.

In our exercise, since the slope was calculated to be 0, the line is horizontal. This means that no matter how much \(x\) changes, \(y\) will always remain constant at the same level.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis. In the slope-intercept form \(y = mx + b\), \(b\) represents this y-intercept. It tells us the value of \(y\) when \(x = 0\). This point is crucial for graphing, as it gives a clear starting point on the y-axis.
To determine the y-intercept from a given equation of a line, you simply identify the constant term \(b\).

• If \(b = 2\), for example, the line crosses the y-axis at \(y = 2\).
• This point is \((0, b)\) on the graph.

In our problem, the equation \(y = 2\) shows that the y-intercept is 2, meaning the line is always crossing the y-axis at this point. Since our slope was 0, every point on the line lies at \(y = 2\). This makes the line not just pass through the y-axis at \((0, 2)\), but every other point on the line shares this y-coordinate.