Problem 9 - Algebra 1

Question

Write an equation in slope-intercept form of the line that passes through the points. $$ (-1,1),(4,5) $$

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form that passes through the given points is \(y = \frac{4}{5}*x + \frac{9}{5}\)

Step 1: Calculate the Slope

The slope of a line passing through the points \((-1,1)\) and \((4,5)\) can be calculated using the formula for the slope between two points, which is:\[m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\]Substitute the given points into this formula:\[m = \frac{5 - 1}{4 - (-1)} = \frac{4}{5}\]

Step 2: Substitute Slope and a Point Into the Slope-Intercept Form

After calculating the slope, you can find the equation of the line in slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. To find \(b\), plug in the values of \(m\), \(x\), and \(y\) from one of the points. Let's take the point \((-1,1)\):\[1 = \frac{4}{5}*(-1) + b\]

Step 3: Solve for b

Solve the above equation for \(b\):\[1 = -\frac{4}{5} + b\]\[b = 1 + \frac{4}{5} = \frac{9}{5}\]

Step 4: Write the Equation of the Line

Substitute the \(m\) and \(b\) values into the slope-intercept form equation to get the final equation:\[y = \frac{4}{5}*x + \frac{9}{5}\]

Key Concepts

Understanding Slope

The slope of a line is a measure of how steep the line is. It represents the change in the vertical position (y) of a line for a specific change in horizontal position (x).
To find the slope between two points on a graph, we use the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]. This is because \(m\) stands for the slope, and the equation calculates the rise over the run between two points.
The points are usually given as \((x_1, y_1)\) and \((x_2, y_2)\). Substitute these values from the given points into the formula to find \(m\).

• If the value of \(m\) is positive, the line slopes upwards from left to right.
• If the value is negative, the line slopes downwards.
• If \(m = 0\), the line is horizontal, indicating no vertical change.

Forming a Linear Equation

A linear equation is an equation that models a straight line on a graph. It is commonly represented in the slope-intercept form, which is \(y = mx + b\).
This specific form highlights two important elements – the slope (\(m\)) and the y-intercept (\(b\)).
Whenever you're asked to find the equation for a line given two points:

• First, calculate the slope using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
• Then, substitute one of the points and the calculated slope into the slope-intercept form to solve for \(b\).
• Finally, write the full equation using your slope and calculated y-intercept.

Remember, the relationship shown by a linear equation remains constant as you move along the line, making it predictable and simple to graph once the slope and y-intercept are known.
Practice helps in becoming quicker and more comfortable with identifying these key components.

The Role of the y-intercept

The y-intercept is another crucial component of a line’s equation. It represents the point where the line crosses the y-axis.
This intercept occurs where the value of \(x\) is 0. In slope-intercept form \(y = mx + b\), \(b\) is the y-intercept.
To find \(b\) when you have the slope and a point on the line:

• Use the form \(y = mx + b\) to plug in the slope and the coordinates of one known point.
• Solve the equation for \(b\), this will give you your y-intercept.

Finding the y-intercept helps to accurately plot the line, as it gives a starting point on the graph.
The unique aspect of the y-intercept is that it provides consistency across different representations of a line and remains unchanged.