Problem 30 - Algebra 1

Question

Write an equation in point-slope form of the line that passes through the given point and has the given slope. $$ (-1,-3), m=4 $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form that passes through the point (-1, -3) and has a slope of 4 is \( y = 4x + 1 \).

Step 1: Identify the given point and slope

The point given is (-1,-3), so x1 is -1 and y1 is -3. The slope given is 4, so m is 4.

Step 2: Substitute the values into the point-slope form of equation

To write the equation, substitute the values of the point and the slope into the point-slope form. The slope m is 4, the x-coordinate x1 is -1 and the y-coordinate y1 is -3. So, the equation becomes \[y - (-3) = 4(x - (-1))\] or equivalently \[y + 3 = 4(x + 1)\].

Step 3: Simplify the equation

By simplifying the equation, we get \[y = 4x + 4 - 3\], which results in \[y = 4x + 1\].

Key Concepts

Equation of a Line

When you're asked to find the equation of a line, it's all about understanding how a line behaves on a coordinate plane and expressing it mathematically. The equation helps depict the relationship between every point that lies on the line.
For the given exercise, we're focusing on the **point-slope form** of a line's equation. This form is perfect when you have a single point and the line's slope.
The point-slope form is expressed as:

• \[y - y_1 = m(x - x_1)\]

Here, - \(m\) represents the slope,- \((x_1, y_1)\) is the point through which the line passes. Finding the equation of the line becomes a matter of substituting these values. Once the values are substituted, you can further simplify the equation to another form, such as the **slope-intercept form** (\(y = mx + b\)). This form is convenient for graphing as it directly tells you the slope and the y-intercept.

Slope

The slope of a line is a numerical value that describes its steepness and direction. Imagine a hill: the more it rises, the steeper it is, similar to how a slope works in coordinate geometry.
There are a few key ideas about slope to keep in mind:

• It's calculated as the 'rise over run,' meaning the vertical change divided by the horizontal change between two points.


• The symbol for slope is usually \(m\).


• A positive slope indicates that the line goes upwards as you move from left to right, while a negative slope implies it goes downwards.


• If the slope is zero, the line is perfectly horizontal, and if it's undefined, the line is vertical.

Understanding slope is crucial because it not only tells you how to draw the line but also gives insight into the relationship between the variables in your equation, like how one changes with respect to the other.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. In this system, a plane is defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
Points on this plane are identified by ordered pairs \((x, y)\), which are essentially the coordinates that describe the exact location of a point.
The beauty of coordinate geometry lies in how it allows us to represent geometric shapes and forms algebraically. Lines, for instance, can be represented by equations, enabling us to solve problems involving shapes algebraically.

• With point-slope form, you can easily determine a line if you know one point and its slope, as shown in the exercise.


• It bridges the gap between algebra and geometry, allowing us to explore geometric concepts with numerical and algebraic methods.

Whether working with lines, curves, or segments, coordinate geometry empowers you to visualize and solve spatial problems with precision and clarity.