1.3 Solving System of Equations by Graphing
Example: Solve the System of Equations by Graphing
Consider the following system of equations:
y = 2x + 4 y = -x + 1
Step 1: Identify the \(y-Intercept\) and Slope
Compare the given equations with the general slope-intercept form of a linear equation, \(y=mx+b\); '\(m\)' is the slope and '\(b\)' is the \(y-intercept\).
For the first equation, the \(y-intercept\) '\(b\)' is 4 and the slope '\(m\)' is 2. This means when \(x = 0\), \(y = 4\), and for every 1 unit increase in \(x\), \(y\) increases by 2 units.
For the second equation, the \(y-intercept\) ‘\(b\)’ is 1 and the slope '\(m\)' is -1. This means when \(x = 0\), y\( = 1\), and for every 1 unit increase in \(x, y\) decreases by 1 unit.
Step 2: Graph Each Equation
Start by plotting the y-intercepts on the \(y-axis\). Then, use the slope to find the next point and draw the line.
For the first equation, start at \((0,4)\) and since the slope is \(2\), for every \(1\) unit you move to the right, move 2 units up.
For the second equation, start at \((0,1)\) and since the slope is \(-1\), for every \(1\) unit you move to the right, move 1 unit down.
Step 3: Find the Intersection Point
The point where the two lines intersect is the solution to the system of equations. In this example, the lines intersect at the point \((-1,2)\).
Step 4: Check Your Solution
Substitute the \(x\) and \(y\) values of the intersection point into both original equations to verify that they are true.
For the first equation: L.H.S =2, and R.H.S.=2×(-1)+4=2 L.H.S.= R.H.S.= 2 For the second equation: L.H.S =2, and R.H.S.=-1×(-1)+1=2 L.H.S.= R.H.S.= 2
Since both equations are true, the solution to the system of equations is \((-1,2)\).