Question 702042: 3x+y=10
y=-3x+4
what is the salution to rhis sytem
a. no solution
b.inf.m.sol
c.1,7
d. -1,7
Answer by RedemptiveMath(80)   (Show Source):
You can put this solution on YOUR website! The answer is no solution. I will attempt to show you how we derive this answer from the given equations.
We are given the equations 3x+y=10 and y=-3x+4. We can treat these two equations as a linear system, and thus we can use the respective operations (elimination, substitution, and graphing).
1. Elimination
y=-3x+10 (subtracted 3x from each side)
y=-3x+4
0 ≠ 6.
I subtracted 3x in the first equation in order to get the common variables on the same sides between both equations. Using elimination, we see that both variables will be cancelled out through subtraction. We are left with 0 = 6, which we know cannot be logical. Therefore, we use the phrase "no solution" in order to state that these two equations have no common ordered pair as a solution.
2. Substitution
y=-3x+10
y=-3x+4
-3x+10=-3x+4
10 ≠ 4.
Using substitution, we can see that the two equations are not equivalent to each other. -3x + 10 and -3x + 4 obviously cannot be the same because one is being changed by 10 and the other by 4 (both are being manipulated by a multiplication factor of -3). We arrive at the final equation 10 = 4, which we know cannot be true. Therefore, we still see that these two equations do not have a similar solution as an ordered pair.
3. Graphing
If we were to graph both of these lines, we'd see that they are parallel to each other. We do not even have to graph the lines to see that they are parallel. Since both of the slopes are -3 in the slope-intercept form, these lines are considered to be parallel. Therefore, the answer is "no solution" because the lines never intersect.
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