A point of intersection between 2 or more equations can be found by either graphing the equations, using the substitution method, or using the elimination method.
Graphing the equations involves plotting both lines on a graph and determining the point in which both lines intersect - it's usually easier to write each equation in slope-intercept form (y=mx+b) before plotting.
The elimination method requires the addition or subtraction of the equations, in order to eliminate a variable. This can usually be achieved by multiplying one or both equations by a certain value. The resulting equation may require to be rearranged so that the variables are in the same order (usually in the form of Ax + By = C). Eliminate fractions or decimals in order to simplify the process. This method also, in part, requires the substitution method in order to work - this is outlined below.
The substitution method involves choosing one of the two given equations and solving for one of the two variables provided (usually [tex]x[/tex] and [tex]y[/tex]). It's best to choose a variable with a coefficient of 1, if that is possible.
Substitute the resulted expression into the other equation and solve for the remaining variable. Once that value is determined, substitute that value into one of the equations in order to determine the second variable.
To solve the equations [tex]2x-3y=8[/tex] and [tex]y=-x+2[/tex], let's use the substitution method.
Equation 1 ⇒ [tex]2x-3y=8[/tex]
Equation 2 ⇒ [tex]y=-x+2[/tex]
Since the second equation ([tex]y=-x+2[/tex]) is already solved for variable [tex]y[/tex], we can substitute this into equation one.
[tex]2x-3y=8[/tex]
[tex]2x-3(-x+2)=8[/tex]
The resulting equation is [tex]2x+3x-6=8[/tex]
Move all the [tex]x[/tex] terms to one side, and all the other terms to the other side.
[tex]5x=14[/tex]
Isolate [tex]x[/tex] by dividing each side by 5
x = 2.8
Since we have now determined the [tex]x[/tex] value, we can substitute this into any one of the two equations.
[tex]y=-x+2[/tex]
[tex]y=-2.8+2[/tex]
After solving for [tex]y[/tex], we can now determine that the resulting value is -0.8
As a result, the x value is 2.8 and the y value is -0.8
Final answer: (2.8, -0.8)
