Problem 21 - Algebra 1

Question

Write an equation of the line that passes through the point and has the given slope. Write the equation in slope-intercept form. $$(0,-2), m=4$$

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y = 4x - 2\).

Step 1: Understand the Slope-Intercept Form

The slope-intercept form of a line is \(y = mx + b\), where m is the slope and b is the y-intercept. We have our slope (m), which is 4.

Step 2: Insert the known point

Now, we need to substitute the values of the point (0, -2) into the equation in place of x and y. So, -2 = 4 * 0 + b.

Step 3: Solve for the y-intercept (b)

We can solve this equation for b (the y-intercept) and we find b = -2.

Step 4: Write the equation of the line

Now, we substitute m and b back into the slope-intercept form of the equation to get our result. The final equation is \(y = 4x - 2\) which represents the line with slope 4 passing through the point (0, -2).

Key Concepts

Writing Linear Equations

Constructing a linear equation can be compared to crafting a navigational guide for a traveler. The traveler is the point on the graph and the compass is the slope, guiding the direction. In our case, we have a point—(0, -2)—and a slope—4. These two pieces of information are the key ingredients to our equation recipe. The slope-intercept form, \( y = mx + b \), is the standard format where 'm' reflects the slope and 'b' the y-intercept, the point where the line crosses the y-axis.

To write the equation, we place the slope, m=4, firmly in our equation. Next, we pinpoint our y-intercept, which will emerge like a beacon from the fog as we insert our given point into the equation. We find out that our line slices through the y-axis at -2, giving us the final equation of \( y = 4x - 2 \). This equation now serves as a roadmap, detailing the slope and starting point of our line.

Slope of a Line

Imagine the slope of a line as the steepness of a hill; the larger the slope, the steeper the hill. Mathematically, slope is defined as the ratio of the rise over the run between any two points on a line. In other words, it shows how much y increases (or decreases) for a one-unit increase in x.

In the equation \( y = mx + b \), 'm' represents this rate of change. If our slope, m, is 4, it means that for every step we take to the right along the x-axis, we must climb 4 steps up or down the y-axis, creating a sharp upward incline if m is positive or a downward plunge if it is negative.

Y-Intercept

The y-intercept is the location where the line crosses the y-axis, providing a starting point or home base for the line on the graph. It's the value of y when x is zero, thus named the 'y-intercept'.

In the equation \( y = mx + b \), the 'b' represents the y-intercept. When we state that the line crosses the point (0, -2), it is clear that at x=0, the equation yields y=-2. This reveals the secret of the y-intercept's identity, said to be the anchor point from which the slope pulls the line upward or downward.

Graphing Linear Equations

Transferring our slope-intercept form equation to a visual representation, we start graphing. First, we anchor the line at its y-intercept: point (0, -2) in this case. From there, we harness the slope to dictate the line's direction. With a slope of 4, or 4/1, we move 4 units up for every unit we move to the right.

This graphing technique breathes life into our equation, turning abstract numbers into a visible, tangible line on the graph. Additionally, graphing serves as a sanity check, validating that our written equation does, in fact, pass through the specified points and has the correct steepness as indicated by the slope.