How do I go about solving the two following systems of linear equations?
1. $$\begin{align} 9x - 17y &= -20\\ -13x + 7y &= -94 \end{align}$$
2. $$\begin{align} 5x - 2y &= -13\\ 4x + 5y &= -6 \end{align}$$
I come with the following wrong answers.
- $x = -\dfrac75, y = 3$
- $x = 11, y = \dfrac5{46}$
Thank you.
The straight forward way without a lot of mathematical background is to eliminate one variable first:
I will do the second problem as an example and leave the first for you:
We will eliminate $x$ by first multiplying both equations with a constant factor such that the coefficient for $x$ take the same value in the multiplied equations. Then we can eliminate $x$ by subtracting them: $$20x-8y=-52$$ $$20x+25y=-30$$
We have multiplied the first equation by $4$ and the second by $5$.
Now subtract them: $$-33y=-22$$
Thus $y=\frac{2}{3}$. Putting this in the second equation (or the first, which one does not matter) we get $4x+5\cdot\frac{2}{3}=-6$ thus $x=-\frac{7}{3}$. We can use the other equation (here the first one) to test that we did not make an error in the calculation: $$-5\cdot\frac{7}{3}-2\cdot\frac{2}{3}=-\frac{35+4}{3}=-13$$ That is correct!
Note that a special case can occur when both equations eliminate to $0=0$ when subtracting. In this case the equations are called "linear dependent" and there is an infinite amount of solution pairs for $\left(x,y\right)$ ...
Remark: If you are serious about learning this, you should try to apply the exact same procedure to the first problem and update your question with your calculations in case any errors occur. You can also make up more problems and test yourself, again updateing your question in case of errors. This is the only way for you (and us) to make sure you have understood the procedure.