Graphing Linear System of Equations
1) Graphing a Line
A line can be graphed by using following three methods:
Note: Linear equation can be in the following forms:
- \(Ax+by+c=0\)
- \(y=mx+b\)
- \(Ax+By=C\)
a) Using a Table of Values
- Choose several values for \(x\).
- Substitute each \(x-value\) into the equation to find the corresponding \(y-value\).
- Plot these \((x,y)\) pairs on the graph.
- Draw a line through the points.
Example: Let’s graph \(y = 2x + 1\) using a table of values.
| x | y |
|---|---|
| -1 | -1 |
| 0 | 1 |
| 1 | 3 |
If \(x=-1\), then \(y=2(-1)+1\)
\(y=-2+1\)
\(y=-1\)
and so on for rest of the \(x-values\).
b) Using Slope and \(y-intercept\)
- Identify the slope (\(m\)) and \(y-intercept\) (\(b\)) from the equation of the form \(y = mx + b\).
- Plot the \(y-intercept\) on the \(y-axis\).
- From the \(y-intercept\), use the slope to find another point. Remember, slope is rise over run.
- Draw a line through the points.
Example: Let’s graph \(y = 2x + 1\) using slope and \(y-intercept\).
Slope (\(m\)) = 2
\(y-intercept\) (\(b\)) = 1
How to change a linear equation from standard form to slope-intercept form?
The standard form of a linear equation is \(Ax + By = C\), where \(A\),\(B\), and \(C\) are constants. The slope-intercept form is \(y = mx + b\), where '\(m\)' is the slope and '\(b\)' is the \(y-intercept\).
To convert from standard form \((Ax + By = C)\) to slope-intercept form \((y = mx + b)\):
- Solve for \(y\): Rearrange the standard form equation to solve for y. This usually involves subtracting '\(Ax\)' from both sides to isolate '\(By\)' on one side, and then dividing every term by \(B\).
- Identify the slope and \(y-intercept\): Once the equation is in slope-intercept form, the coefficient of \(x\) is the slope (\(m\)), and the constant term is the y-intercept (\(b\)).
Example: Let’s convert the standard form equation \(2x + 3y = 6\) to slope-intercept form.
Subtract \(2x\) from both sides: \(3y = -2x + 6\).
Divide every term by 3: \(y = \frac{-2}{3} x + 2\).
So, the slope-intercept form of the equation \(2x + 3y = 6\) is \(y = \frac{-2}{3} x + 2\).
c) Using \(x\) and \(y-intercepts\)
- To find the x-intercept, set \(y = 0\) and solve for \(x\).
- To find the \(y-intercept\), set \(x = 0\) and solve for \(y\).
- Plot these intercepts on the graph.
- Draw a line through the points.
Example: Let’s graph \(2x + y = 2\) using \(x\) and \(y-intercepts\).
\(x-intercept\): Set \(y = 0\), solve for \(x: x = 1\).
\(y-intercept\): Set \(x = 0\), solve for \(y: y = 2\).
2) Solving a System of Equations by Graphing
- Graph each equation on the same set of axes.
- The point(s) where the lines intersect is the solution to the system.
Example: Let’s solve the system of equations \(y = 2x + 1\) and \(y = -x + 4\) by graphing.
Graph \(y=2x+1\) and \(y=-x+4\) on same set of axes.
The lines intersect at the point (1, 3), so this is the solution to the system.
