## Question

###### How do you find the point of intersection for #x-4y=6# and #3x+4y=10#?

## Answers

The unique answer is the point

#(x,y)=(4,-1/2)# ## Explanation:

Method 1 (Substitution):One way to solve this is to take the first equation of the system,#x-4y=6# , and solve it for#x# in terms of#y# to get#x=4y+6# . Then plug this into the second equation to make#3x+4y=10# become#3(4y+6)+4y=10# , which simplifies to#12y+18+4y=10# , then#16y=-8# , then#y=-1/2# .Now use the equation

#x=4y+6# to find#x# :#x=4*(-1/2)+6=-2+6=4# .The answer is the point

#(x,y)=(4,-1/2)# .(BTW, all this work implies the only possible solution is

#(x,y)=(4,-1/2)# (so if it is a solution, it is unique). You can confirm it truly is a solution by substitution back into the original system (checking your work proves it is a solution).)

Method 2 (Elimination):Notice that the#-4y# in the first equation and the#4y# in the second equation will cancel if we add the two equations. Do so to get:#4x=16# so#x=4# . Then substitute this into either of the original equations to get, for instance#4-4y=6# so#4y=-2# and#y=-1/2# , resulting in#(x,y)=(4,-1/2)# .A picture of this situation is shown below. The equation

#x-4y=6# is equivalent to#y=1/4 x-3/2# and is the red line with a slope of#1/4# and a#y# -intercept of#-3/2# . The equation#3x+4y=10# is equivalent to#y=-3/4x+5/2# and is the blue line with a slope of#-3/4# and a#y# -intercept of#5/2# .