Section 5.4
Intersections of Lines by Substitution
When we have two linear equations in $x$ and $y$, each represents a line in a coordinate system. Typically, these lines intersect in a single point (the other possibilities are that the lines are parallel, and therefore don't intersect; or that they are in fact the same line in disguise, and so intersect in all their points).
One way to find the point of intersection of two lines is by substitution. This is recommended only when one or the other of the equations can be solved easily for either $x$ or $y$.
Examples:
a. $\left\{ \,\begin{matrix}
2x+3y=4 \\
3x+y=-1 \\
\end{matrix} \right.$
Solve the second equation for $y$, and substitute that expression for $y$ in the first equation.
$\begin{align}
& y=-3x-1 \\
& 2x+3(-3x-1)=4 \\
& 2x-9x-3=4 \\
& -7x=7 \\
& x=-1 \\
& y=-3x-1=-3(-1)-1=2
\end{align}$
The point of intersection of the lines is $(-1,2)$.
b. $\left\{ \,\begin{matrix}
-x-3y=4 \\
4x-5y=18 \\
\end{matrix} \right.$
Solve the first equation for $x$ and substitute that expression for $x$ in the second equation.
$\begin{align}
& -3y-4=x \\
& 4(-3y-4)-5y=18 \\
& -12y-16-5y=18 \\
& -17y=34 \\
& y=-2 \\
& x=-3y-4=-3(-2)-4=2
\end{align}$
The point of intersection of the lines is $(2,-2)$.
Solving linear systems by substitution: Solving Linear Systems by Substitution
Exercises:
Solve each of the following systems of equations for $x$ and $y$ by substitution.
1. $\left\{ \,\begin{matrix}
2x+y=5 \\
3x-2y=11 \\
\end{matrix} \right.$ 2. $\left\{ \,\begin{matrix}
x+y=12 \\
x-y=30 \\
\end{matrix} \right.$
3. $\left\{ \,\begin{matrix}
4x-y=1 \\
2x+3y=4 \\
\end{matrix} \right.$ 4. $\left\{ \,\begin{matrix}
x-y=-3 \\
4x+3y=2 \\
\end{matrix} \right.$
5. $\left\{ \,\begin{matrix}
8x-3y=42 \\
x+4y=14 \\
\end{matrix} \right.$ 6. $\left\{ \,\begin{matrix}
8x-5y=13 \\
5x+y=4 \\
\end{matrix} \right.$
7. $\left\{ \,\begin{matrix}
3x-y=5 \\
2x+3y=18 \\
\end{matrix} \right.$ 8. $\left\{ \,\begin{matrix}
x-5y=4 \\
-2x+10y=-10 \\
\end{matrix} \right.$
9. $\left\{ \,\begin{matrix}
3x+y=4 \\
-2x+3y=-21 \\
\end{matrix} \right.$ 10. $\left\{ \,\begin{matrix}
3x+4y=7 \\
x+2y=2 \\
\end{matrix} \right.$
11. $\left\{ \,\begin{matrix}
3x+4y=9 \\
-x+2y=2 \\
\end{matrix} \right.$ 12. $\left\{ \,\begin{matrix}
4x+3y=-7 \\
x+6y=0 \\
\end{matrix} \right.$
13. $\left\{ \,\begin{matrix}
11x+2y=-7 \\
x+6y=11 \\
\end{matrix} \right.$ 14. $\left\{ \,\begin{matrix}
7x+2y=20 \\
-x+y=1 \\
\end{matrix} \right.$
15. $\left\{ \,\begin{matrix}
2x+y=24 \\
3x-5y=-3 \\
\end{matrix} \right.$ 16. $\left\{ \,\begin{matrix}
3x-y=4 \\
6x+y=-1 \\
\end{matrix} \right.$
17. $\left\{ \,\begin{matrix}
y=\tfrac{2}{3}x-3 \\
y=5x+1 \\
\end{matrix} \right.$ 18. $\left\{ \,\begin{matrix}
y=2x-3 \\
3x-4y=-8 \\
\end{matrix} \right.$
19. $\left\{ \,\begin{matrix}
x+4y=1 \\
3x-2y=-4 \\
\end{matrix} \right.$ 20. $\left\{ \,\begin{matrix}
2x+y=7 \\
x-4y=-1 \\
\end{matrix} \right.$
21. $\left\{ \,\begin{matrix}
\tfrac{x}{3}+\tfrac{y}{2}=7 \\
x+y=15 \\
\end{matrix} \right.$ 22. $\left\{ \,\begin{matrix}
y-4x=-10 \\
3x-\tfrac{3}{2}y=0 \\
\end{matrix} \right.$
23. $\left\{ \,\begin{matrix}
\tfrac{2x}{5}-\tfrac{y}{2}=7 \\
x-4y=1 \\
\end{matrix} \right.$ 24. $\left\{ \,\begin{matrix}
2x+5y=8 \\
x+\tfrac{2}{3}y=\tfrac{1}{3} \\
\end{matrix} \right.$
25. $\left\{ \,\begin{matrix}
\tfrac{5x}{3}-\tfrac{2y}{5}=3 \\
2x-y=1 \\
\end{matrix} \right.$ 26. $\left\{ \,\begin{matrix}
\tfrac{4x}{3}+\tfrac{3y}{4}=7 \\
x-3y=10 \\
\end{matrix} \right.$
27. $\left\{ \,\begin{matrix}
x=3y-7 \\
y=2x-1 \\
\end{matrix} \right.$ 28. $\left\{ \,\begin{matrix}
y=4x+10 \\
x=2y-13 \\
\end{matrix} \right.$