Slope-Intercept Form
Understanding the slope-intercept form of a linear equation is essential for graphing linear systems. It is represented as \( y = mx + b \), where \( m \) stands for the slope, and \( b \) indicates the y-intercept, the point where the line crosses the y-axis.
The slope, \( m \), expresses how steep the line is, meaning how much y changes for a one-unit change in x. If the slope is positive, the line slants upwards from left to right. Conversely, if the slope is negative, the line tilts downwards. The intercept, \( b \), gives a starting point to draw the line.
For the given exercise, we have the equations \( -x + y = -2 \) and \( 2x + y = 10 \) which convert to \( y = x - 2 \) and \( y = -2x + 10 \) respectively in slope-intercept form. This transformation simplifies graphing since you can directly plot the y-intercept and use the slope to find additional points.
Systems of Equations
A system of equations consists of two or more equations with the same variables. The goal is to find a set of values for the variables that satisfies all equations simultaneously. Systems can be solved using various methods: graphically, substitution, elimination, or matrix operations.
When solving graphically, which is the method used in our exercise, you represent each equation as a line on the same graph. The point where the lines intersect represents the solution to the system, providing the values for x and y that work for both equations. In our case, visual analysis of the lines described by the equations \( y = x - 2 \) and \( y = -2x + 10 \) would reveal if they intersect, and where, to determine the validity of a given solution like \( (4,-2) \).
Plotting Equations
To plot equations on a graph, you'll need to identify points that satisfy the equation and then connect these points to visually represent the equation as a line on the graph. The slope-intercept form simplifies this process by providing a clear starting point, the y-intercept \( b \), and a pattern for identifying additional points through the slope \( m \).
For instance, using the slope-intercept form, if the slope is 2, you'll rise 2 units up for every one unit you move to the right from the y-intercept. For a negative slope like -2, you'll move down two units for every one unit you move to the right. After plotting enough points, draw a line through all of them to represent the equation graphically.
Ordered Pairs
An ordered pair \( (x, y) \) denotes the coordinates of a point on a two-dimensional graph. The first number, x, indicates the position along the horizontal axis, while the second number, y, represents the vertical axis position. When dealing with linear systems, an ordered pair is a solution if it satisfies all the equations simultaneously.
In the context of the exercise, after graphing the lines for each equation, we check if the ordered pair \( (4,-2) \) lies at the intersection of these two lines. If it does, it means \( (4,-2) \) is a solution to the system. Otherwise, it's not. Concluding whether an ordered pair is a solution primarily relies on verifying if it fulfills both the equations involved.
