Section 5.5
Intersections of Lines by Elimination
A second method of solving two equations in two unknowns simultaneously for the unknowns is by the method of linear combinations. This method is particularly appropriate when solving an equation for one of the unknowns will necessarily introduce fractions into the work. The idea is to multiply the equations by specific constants so that the coefficients of one or the other of the variables are opposites.
Examples:
a. $\left\{ \,\begin{matrix}
3x+2y=1 \\
6x+5y=4 \\
\end{matrix} \right.$
Multiply the first equation by$-2$.
$\left\{ \,\begin{matrix}
-6x-4y=-2 \\
6x+5y=4 \\
\end{matrix} \right.$
Now when the equations are added together (using the Addition Property of Equality), the terms with the $x$ drop out, giving $y=2$. Then substitution of that value into either of the original equations leads to $x=-1$. The point of intersection of the two lines is $(-1,2)$.
b. $\left\{ \,\begin{matrix}
7x-4y=3 \\
6x+5y=11 \\
\end{matrix} \right.$
Multiply the top equation by 5 and the bottom equation by 4; that will make the coefficients of y in the two equations be $-20$ and $20$.
$\left\{ \,\begin{matrix}
35x-20y=15 \\
24x+20y=44 \\
\end{matrix} \right.$
Adding the two equations produces the result $59x=59$, so $x=1$. Substituting this value for x in the first equation then gives $y=1$. The point of intersection of the two lines is $(1,1)$.
c. $\left\{ \,\begin{matrix}
7x+5y=15 \\
4x+2y=7 \\
\end{matrix} \right.$
Multiply the top equation by 2 and the bottom equation by $-5$.
$\left\{ \,\begin{matrix}
14x+10y=30 \\
-20x-10y=-35 \\
\end{matrix} \right.$
Adding the two equations gives $-6x=-5$, so $x=\frac{5}{6}$. This can be substituted into one of the original equations to obtain the value of y. However, that introduces a lot of fraction work. Another possibility is to form a different combination of the original two equations in order to get the variable x to drop out.
Multiply the top equation by 4 and the bottom equation by $-7$.
$\left\{ \,\begin{matrix}
28x+20y=60 \\
-28x-14y=-49 \\
\end{matrix} \right.$
Adding these gives $6y=11$, so $y=\dfrac{11}{6}$.
Solving systems of equations by elimination: Solving Systems of Equations by Elimination
Exercises:
Solve each of the following systems of equations for $x$ and $y$ by forming linear combinations.
1. $\left\{ \,\begin{matrix}
4x+7y=10 \\
2x+5y=8 \\
\end{matrix} \right.$ 2. $\left\{ \,\begin{matrix}
6x+5y=16 \\
-3x+4y=5 \\
\end{matrix} \right.$
3. $\left\{ \,\begin{matrix}
6x+2y=11 \\
3x-4y=8 \\
\end{matrix} \right.$ 4. $\left\{ \,\begin{matrix}
2x+7y=3 \\
6x-3y=-15 \\
\end{matrix} \right.$
5. $\left\{ \,\begin{matrix}
5x-3y=19 \\
6x+5y=-3 \\
\end{matrix} \right.$ 6. $\left\{ \,\begin{matrix}
2x-6y=1 \\
5x+4y=12 \\
\end{matrix} \right.$
7. $\left\{ \,\begin{matrix}
5x-3y=41 \\
3x+6y=9 \\
\end{matrix} \right.$ 8. $\left\{ \,\begin{matrix}
3x-4y=20 \\
5x+2y=16 \\
\end{matrix} \right.$
9. $\left\{ \,\begin{matrix}
5x+2y=3 \\
2x+5y=4 \\
\end{matrix} \right.$ 10. $\left\{ \,\begin{matrix}
3x+y=7 \\
2x-5y=16 \\
\end{matrix} \right.$
11. $\left\{ \,\begin{matrix}
x+7y=7 \\
3x+y=11 \\
\end{matrix} \right.$ 12. $\left\{ \,\begin{matrix}
2x-5y=31 \\
3x+y=4 \\
\end{matrix} \right.$
13. $\left\{ \,\begin{matrix}
3x-7y=5 \\
5x+2y=22 \\
\end{matrix} \right.$ 14. $\left\{ \,\begin{matrix}
2x+5y=8 \\
11x+7y=3 \\
\end{matrix} \right.$
15. $\left\{ \,\begin{matrix}
8x-5y=3 \\
3x+7y=-61 \\
\end{matrix} \right.$ 16. $\left\{ \,\begin{matrix}
2x-3y=5 \\
3x-7y=15 \\
\end{matrix} \right.$
17. $\left\{ \,\begin{matrix}
x-2y=-1 \\
3x+2y=10 \\
\end{matrix} \right.$ 18. $\left\{ \,\begin{matrix}
2x+3y=6 \\
5x-4y=20 \\
\end{matrix} \right.$
19. $\left\{ \,\begin{matrix}
3x-4y=5 \\
4x-7y=5 \\
\end{matrix} \right.$ 20. $\left\{ \,\begin{matrix}
3x+4y=-3 \\
2x-3y=32 \\
\end{matrix} \right.$
21. $\left\{ \,\begin{matrix}
6x+5y=7 \\
2x-3y=7 \\
\end{matrix} \right.$ 22. $\left\{ \,\begin{matrix}
5x+6y=7 \\
3x+4y=6 \\
\end{matrix} \right.$
23. $\left\{ \,\begin{matrix}
7x+10y=7 \\
3x+4y=6 \\
\end{matrix} \right.$ 24. $\left\{ \,\begin{matrix}
3x+4y=7 \\
5x+7y=6 \\
\end{matrix} \right.$
25. $\left\{ \,\begin{matrix}
2x+4y=7 \\
9x+7y=6 \\
\end{matrix} \right.$ 26. $\left\{ \begin{matrix}
2x-5y=7 \\
7x+3y=4 \\
\end{matrix} \right.$
27. $\left\{ \,\begin{matrix}
2x-3y=7 \\
9x+5y=6 \\
\end{matrix} \right.$ 28. $\left\{ \begin{matrix}
3x-7y=7 \\
7x+3y=4 \\
\end{matrix} \right.$