Problem 62 - Algebra 1

Question

Write an equation of the line in slope-intercept form that passes through the two points, or passes through the point and has the given slope. $$(4,-5),(-1,-3)$$

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form that passes through the points (4,-5) and (-1,-3) is \(y = -0.4x - 3.4\).

Step 1: Calculate the slope

The slope (m) of a line passing through the coordinates \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula \(m = (y_2 - y_1) / (x_2 - x_1)\). For the points given in the exercise, which are \((4, -5)\) and \((-1, -3)\), the slope can be calculated as \(m = (-3 - (-5)) / (-1 - 4) = 2 / -5 = -0.4\).

Step 2: Calculate the y-intercept

Now the y-intercept (b) can be calculated. From the slope-intercept equation \(y = mx + b\), we get \(b = y - mx\). Substituting one of the points and the calculated slope, \(b = -5 - (-0.4 * 4) = -5 + 1.6 = -3.4\).

Step 3: Write the final equation

By substituting the slope and y-intercept values into the slope-intercept form we get the equation of the line, \(y = -0.4x - 3.4\).

Key Concepts

Equation of a Line

Understanding the equation of a line is essential in algebra. The slope-intercept form is the most popular form and is expressed as \( y = mx + b \). Here, \( m \) represents the slope of the line, which measures how steep the line is. Meanwhile, \( b \) is the y-intercept, indicating where the line crosses the y-axis. This form is straightforward and immediately shows both the slope and the y-intercept.

When you have two points or a point and a slope, you can easily find a line's equation using this method. Simply insert the slope and solve for the y-intercept to get the final line equation.

Using slope-intercept form allows you to graph lines quickly and see how different equations compare.

Slope Calculation

The slope of a line tells us how steeply it ascends or descends as you move along it. To calculate the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\), you use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

In this exercise, the points given are \((4, -5)\) and \((-1, -3)\). Plugging into the formula:

• The change in y-values: \(-3 - (-5) = 2\)
• The change in x-values: \(-1 - 4 = -5\)

Therefore, the slope \( m = \frac{2}{-5} = -0.4 \).

This tells us the line decreases as it moves from left to right, indicating a downhill slope.

Y-Intercept

The y-intercept is where a line crosses the y-axis. This is crucial because it shows the starting point of the line when \( x = 0 \). Once you know the slope, you can find the y-intercept using one of the points and the slope formula.

Say we use the point \((4, -5)\) and the slope \(-0.4\). We substitute into the slope-intercept formula \( y = mx + b \), solving for \( b \):

• Insert \( -5 \) for \( y \)
• Use \( -0.4 \times 4 \) which is \(-1.6\)
• So, \( b = -5 - (-1.6) = -3.4 \)

Hence, the y-intercept is \(-3.4\). In the final line equation \( y = -0.4x - 3.4 \), this confirms the line's vertical position on the graph.