Pre-Calculus - Honors
Lesson 95
Arn; How do we solve a system of equations by elimination?
Dp*"NgW; 1) Solve for x and y algebraically:
n
4x+5y=23
lx -y =5
pgvglqpmgn!: >How can we add these equations so that one variables
can get cancelled?
<We can multiply one of the equations so that one variable can
become an opposite - we have two options to do that:
Here's one: 4x + 5y = 23
-4x + y = -5 Notice that the entire equation has
6y = 18 its signs changed so that the 4x
y=
has an opposite. We can then add
and solve for y. From there we
solve for x and check.
3
4x+5(3)=23
4x+15=23
4x=8
x=2
So (x,y)=(2,3)
We won't check now let's try another method:
4x+5y=23
25
24x 48
x =)
,
20x - 5v =
--
Here we multiply the entire second
equation by 5 so that the 5y gets an
opposite and we solve for x first.
Ur/e then can solve for y and check.
4(2)+5Y=23
8+5y=23
5Y
= 15
3
y=
Once again (x, y) = (2,3)
We see that we get the same solution for (x, y ) either way.
Now we can check: aQ) + 5(3) =
4(2) - 3 = 5
23
23 =23
8 + 15 =23 "
2) Solve algebraically:
3x+y=6
8 -3=5
5=5
x + 3y = 10
(x,y)=(1,3)
3x+y=6
"
-3x-9y=-30 3x+3=6 ch: 3(1)+(3)=6 (1)+a13;=19
-8y=-24 3x=3
3 + 3 =6 1+9 =10
. 6 =6
y=3
x=1
10 =10
There are other ways of doing this.
Apdteattqns"; Solve algebraically:
3) 3x
+2y=4 4) x-2y=3 5) 2x-y- I
-2x+4y=1 4x-2y=2
5x-2y=3
Pre-Calculus - Honors
Ans:vslsi
3) 3x+2y=4
Lesson 95
912+2Y=4
2y=812-912
5x-2v=8
8x= 12
x = 312
2y = -Y,
y = _Tu (x, y) = (312, _%)
Graphically we see this as two lines that intersect at one point.
Check this on your calculator.
{
4) 2(x-2y=31 ) 2x-4Y=6
-2x+ 4V = 1
-2x+ 4Y = 1
0=T
(x,y)={
}"
We say the system is inconsistent because A = 7.
Graphically we see this as two lines that never intersect (which means,
of courSe, that they are parallel). This is a case where there are
simply NO numbers that can work. Check this on your calculator.
5)
)
4X-2Y=2
-2(2x-
1Y
= 1)
-4x + 2Y = -2
4x-2V=2
0=0
(x,y)={ }
Here we say that the equations are equivalent because 0 = 0.
Graphically we see this as two lines that are identical (or non-distinct)
because one lies on top of the other (or the occupy the same space).
This is a case where there are TOO MANY numbers - an infinite
number in fact - that will work. Check this on your calculator.
Hs_=meu rk; HEATH; p543: #14, 15, 16, 21,23,25
3'd ed. HOUGHTON-MIFFLIN: p505: #14, 15, 16, 23,27,29