1.) y = x – 4;
y = 3x + 2.
Subtract the first equation from the second:
2x + 6 = 0.
Subtract 6 from both sides:
2x = –6.
Divide both sides by 2:
x = –3.
Substitute this into the first equation, above:
y = –3 – 4.
Combine terms: y = –7.
Now, to check your answer, substitute the solution into the two equations given:
–7 = –3 – 4.
–7 = 3(–3) + 2 = –9 + 2.
Both of these statements are true, and, so, we conclude that we have found the correct solution, namely: x = –3, y = –7.
If you graph the two equations that were given, you will find that they cross at the point, (–3, –7).
2.) 2x – y = 8;
x + 2y = 9.
This is only slightly more difficult. Multiply the second equation by 2.
2x + 4y = 18.
Subtract the first equation from this last result. Notice that, since the two equations we are subtracting now have the same co-efficient for x (namely, 2), this will eliminate x from the resulting equation:
5y = 10.
Now, divide both sides by 5:
y = 2.
Now, substitute this into either equation:
2x – 2 = 8.
Add 2 to both sides:
2x =10.
Halve both sides: x = 5.
So, our answer is x = 5, y = 2.
Let’s check it, now, by substituting this into the original equations:
2(5) – 2 = 10 – 2 = 8.
5 + 2(2) = 5 + 4 = 9.
Both of these are correct; so we must have found the correct solution.
If you graph the two equations that were given, you will find that they cross at the point, (5, 2).
3.) 3x + y = 4;
–6x – 2y = 12.
Now, this one is a little different, and I will show you why.
Following usual procedure, we seek to transform one or both equations, multiplying by some constant, so that either x or y has the same co-efficient in both equations. We do this, so that, when we subtract the two equations, thus transformed, we eliminate one variable, either x or y, whichever we chose. But there is a problem here:
Multiply the first equation by –2. Now, we have:
–6x – 2y = –8;
–6x – 2y = 12.
Now, both statements can not be true: subtract one from the other, and you get:
0 = 20, which simply can not be; we call it a contradiction.
In other words, try as you might, you can not find a value for x and one for y which could ever make the original pair of equations true.
Hence, there is no possible solution for this problem.
Put in terms of graphs, if you make a graph of both of the given equations, you will find that they form two parallel lines that cross nowhere. There can be no point that lies on both lines. Therefore, there is no value for x and another value for y that satisfies both equations, and there can be no solution to the problem, as stated.