Problem 53 - Algebra 2

Question

Write an equation in slope-intercept form for the line that satisfies each set of conditions. passes through \((0,5)\) and \((10,10)\)

Step-by-Step Solution

Answer

The equation is \( y = \frac{1}{2}x + 5 \).

Step 1: Understand Slope-Intercept Form

The slope-intercept form of a line is written as \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.

Step 2: Calculate the Slope

The slope \( m \) is calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substitute the points \((0,5)\) and \((10,10)\) into this formula:\[ m = \frac{10 - 5}{10 - 0} = \frac{5}{10} = \frac{1}{2} \]

Step 3: Identify the Y-Intercept

The point \((0,5)\) provides the y-intercept directly. This means when \( x = 0 \), \( y = 5 \), so \( b = 5 \).

Step 4: Substitute into Slope-Intercept Form

Now that we have \( m = \frac{1}{2} \) and \( b = 5 \), substitute these into the slope-intercept form equation:\[ y = \frac{1}{2}x + 5 \]

Key Concepts

Slope Calculation

Finding the slope of a line is essential for understanding its direction and steepness. It's like figuring out how steep a hill is when you're deciding whether to walk or roll down. For any two points on a line, the slope, represented as \( m \), shows the "rise" over the "run". This means we look at how much the line goes up or down (the rise) for a certain distance across (the run).

The mathematical formula to calculate the slope is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). So what do we do with it? What if we have points

• \((0,5)\)
• \((10,10)\)

By plugging these points into the formula, we find:
\[ m = \frac{10 - 5}{10 - 0} = \frac{5}{10} = \frac{1}{2} \]
This tells us the line rises half a unit for every unit it runs along the x-axis. Now, whenever you see two points, you can measure the tilt of the line like a pro!

Y-Intercept

When you hear 'y-intercept', think of it as the point where a line crosses the y-axis. This is essentially the line's starting point when your trip on the x-axis hasn’t started yet (when \( x = 0 \)).

In our problem, one of the easiest parts is identifying the y-intercept because it can be taken straight from the point \((0,5)\). Notice how the x-value is zero here? That means the line started directly at \(y = 5\) on the y-axis.

So, for our given line, the y-intercept, represented as \( b \), is at the point 5. It's like the starting height of the line when you stand directly on the y-axis. Recognizing the y-intercept is crucial because it tells us exactly where the line sits in relation to our grid when you haven’t moved left or right yet.

Equation of a Line

Creating the equation of a line in slope-intercept form means putting all pieces together. With our slope \( m \) and y-intercept \( b \), we're ready to express the line's behavior using a simple equation formula: \( y = mx + b \).

This structure lets you calculate y-values for any given x along the line. You only need to plug the known slope and y-intercept into this formula. In our example:

• The slope \( m \) is \( \frac{1}{2} \)
• The y-intercept \( b \) is 5

Therefore, the equation becomes:
\[ y = \frac{1}{2}x + 5 \]
This equation shows us everything we need. It tells us how the line tilts and where it crosses the y-axis. For a learner, understanding this form simplifies predicting or plotting any part of the line. It's your tool for navigating the line's behavior on a graph.