Problem 20 - 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. $$(0,2), m=3$$

Step-by-Step Solution

Answer

The equation of the line is \(y = 3x +2\).

Step 1: Identify the slope and point coordinates

First, identify the slope and coordinates of the given point. The slope (\(m\)) is given as 3. The coordinates (\(x,y\)) of the point are provided as (0, 2).

Step 2: Substitute into slope-intercept form

Substituting the slope (\(m\)) and point coordinates (\(x,y\)) into the slope-intercept formula \(y = mx + c\), we get \(2 = 3*0 + c\).

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

Solve the equation from Step 2 to find the y-intercept (\(c\)). In this case, \(c = 2\).

Step 4: Write the equation of the line

Finally, substitute the slope (\(m\)) and y-intercept (\(c\)) into the slope-intercept formula to get the equation of the line. The equation is \(y = 3x + 2\).

Key Concepts

Equation of a Line

Understanding how to write the equation of a line is an essential skill in algebra. In most cases, when we talk about the equation of a line, we refer to the slope-intercept form, which is expressed as \(y = mx + c\). This equation gives a straight line when plotted on a graph.

• Here, \(m\) is the slope, which indicates how steep the line is.
• \(c\) is the y-intercept, which shows where the line crosses the y-axis.

To find the equation of a line when given a point and a slope, you substitute these values into the slope-intercept form and solve for \(c\). This process helps in quickly identifying the characteristics and position of the line on a coordinate plane.

Coordinate Geometry

In coordinate geometry, every point on the plane is represented by a pair of numbers \((x, y)\), also known as coordinates. This system helps in visualizing algebraic expressions geometrically.
The coordinate plane is divided into four quadrants, with the horizontal line (x-axis) and vertical line (y-axis) intersecting at the origin \((0,0)\).

• The slope \(m\) gives the amount by which \(y\) changes for a change in \(x\).
• This makes coordinate geometry a powerful tool for solving geometric problems algebraically.

By using equations like \(y = mx + c\), you can easily draw lines and identify their properties such as intercepts and slopes on this plane. Each equation corresponds to a single line, providing clarity to geometric problems.

Y-Intercept

The y-intercept of a line is a point where the line crosses the y-axis. In the slope-intercept form equation \(y = mx + c\), the y-intercept is represented by the constant \(c\). This point always has an x-coordinate of zero because it's where the line meets the vertical axis.

• To determine the y-intercept from a given point and slope, substitute the point into the equation \(y = mx + c\).
• Then solve for \(c\), as shown in the step-by-step process where \(c = 2\).

Understanding the y-intercept is crucial because it provides a fixed point on the line, making it easier to graph and understand the relationship the line models. This intercept is key in quickly sketching the graph of the equation and analyzing how the line behaves relative to the axes.