Problem 17 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line described below. $$ m=\frac{2}{5}, b=7 $$

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y = \frac{2}{5}x + 7\).

Step 1: Identify the values of slope and y-intercept

The slope 'm' is given as \(\frac{2}{5}\) and the y-intercept 'b' is given as \(7\).

Step 2: Plug the values into the slope-intercept formula

The slope-intercept form of the equation of a line is \(y = mx + b\). Plugging the given values into this equation gives \(y = \frac{2}{5}x + 7\).

Key Concepts

Equation of a Line

The equation of a line is a way to represent all the points that lie on a straight line in a coordinate plane. In algebra, we often use various forms of equations to express lines. One of the most common and useful forms is the slope-intercept form. This form is particularly easy to use because it gives direct insights into the line's slope and y-intercept.
- In mathematics, an equation can be thought of as a rule that every point on the line follows.
- For a linear equation, each point \((x, y)\) on the line satisfies the given equation.Understanding the equation of a line helps us to effectively plot and analyze linear graphs. This knowledge is critical in fields ranging from engineering to economics, where plotting and interpreting data is essential.

Slope

The slope of a line measures how steep the line is. Mathematically, slope is the ratio of the vertical change to the horizontal change between two points on the line.
This is often phrased as "rise over run." The slope formula is typically represented as: \[m = \frac{rise}{run} = \frac{y_2 - y_1}{x_2 - x_1}\]

• If the value of the slope \(m\) is positive, this means the line inclines upward as you move from left to right.
• Conversely, if \(m\) is negative, the line declines or goes downwards.
• A slope of zero implies a horizontal line, and if the slope is undefined, the line is vertical.

Knowing the slope is crucial when sketching a line on a graph, predicting values along the line, and understanding the relationship between the variables being graphed.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis of the Cartesian plane. When a line intersects with the y-axis, it means that the x-coordinate is zero at that particular point. This makes the y-intercept an important feature of the line since it helps in determining where the line lies on the graph.

In the slope-intercept formula \(y = mx + b\), the value of \(b\) represents the y-intercept. This means that if you substitute \(x = 0\) into the equation, what you're left with is \(y = b\). Therefore, the y-intercept can always be quickly spotted in the form of the equation, helping us situate the line rapidly on the graph.
Understanding the y-intercept allows for quick insights into how a line functions and where within the coordinate system it rests, assisting in data interpretation and graphical predictions.