Linear Equations
Linear equations are the simplest form of equations one encounters in algebra. They have the highest power of the variable as one, which causes the graph to be a straight line, hence the term 'linear'. A standard form of a linear equation is represented as ax + by = c, with a, b, and c being constants. In the case of our exercise, we can see that the equation 3x + 2y = 10 fits this definition, with a = 3, b = 2, and c = 10.
To find key characteristics of the line, such as slope and y-intercept, we usually want to manipulate this standard form into the slope-intercept form y = mx + b, where m is the slope and b is the y-intercept. This form instantaneously gives us valuable information about the line without having to perform any complex calculations.
Slope of a Line
The slope of a line is a measure of how steep the line is. It's represented by the letter m and can be thought of as the 'rise over run', or the vertical change divided by the horizontal change between two points on a line. Algebraically, if you have two points (x1, y1) and (x2, y2), the slope is calculated as (y2 - y1)/(x2 - x1).
In our example, after rearranging the linear equation into the slope-intercept form, the slope is directly seen before the x variable. Specifically, for the equation y = -3/2x + 5, the slope m = -3/2. This negative fraction tells us that for every 2 units you move to the right along the x-axis, the line falls by 3 units vertically, making the line descending from left to right.
Y-intercept
The y-intercept of a line refers to the point where the line crosses the y-axis. It is always in the form of (0, b) and can be found by seeing where a line will be when x=0. The y-intercept gives us an initial value or starting point for the line.
In the context of the slope-intercept form y = mx + b, the y-intercept is represented by the b term. Looking back at our sample equation, once it has been manipulated into the form y = -3/2x + 5, it's clear that the y-intercept is the constant 5. This means that the line crosses the y-axis 5 units above the origin.
Algebraic Manipulation
Algebraic manipulation involves using mathematical operations to rearrange and solve equations, making it easier to extract information, like the slope and y-intercept from a linear equation. Key operations include adding, subtracting, multiplying, and dividing terms, and can also involve factoring, expanding, and simplifying expressions.
In our exercise, we performed algebraic manipulation by first subtracting 3x from both sides to isolate terms containing y, and then divided every term by 2 to solve for y. This transformation is essential for converting the equation into the slope-intercept form, allowing us to quickly identify the slope and y-intercept of the line.
