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Chapter 7
Writing Linear Equations
– 5x – 12
5
x 3
5 4
4
–3
– 3x – 20
3
x5
4
3
4
5
–4 2
–4
2
2
–4
2
– 20 8
– 12 =
–4 2
–4 2
-4
–4
2
– 12 8
– 20 =
4 y  3x  4 2 y   x  8
3
1
y  x 1
y
x4
4
2
2 y  5 x  4
5
y
x2
2
2 y  9 x  12
9
y
x6
2
(-5,7)
(-2,-5)
5 y  2 x  25
3 y  7 x  6
2
y
x5
5
7
y  x2
3
Y
a coefficient of 1
1
C
a coefficient of 1
2
– 3y – 1
– 3y – 1
– 3y – 1
– 6y – 2
– 11y – 2
– 11y
– 11
1
1
– 3y – 1
– 3(1) – 1
–4
1
–4
–4
1
–4
1
–8
–4
5
1
–4
– 13 =
–4
3
–1 =
1
Isolate x in the first equation
Because It is easy to do so and
the value is easy to substitute
in the other equation.
Isolate y in the second
Equation because it is easier to
Do so than to isolate y in the
First equation or x in either
equation
x – 5(x – 1) = – 15
x – 5x+5 = -15
– 4x= – 20
x=5
3x+2(– 5x+3)= – 8
3x-10x+6= – 8
– 7x = – 14
x=2
y=5 – 1
y=4
y= – 5(2)+3
y=-10+3
y= – 7
Answer: (2, – 7)
Answer: (5,4)
–3
12x 0
x 0
0
x
0
Y –3
0 –3
x
y
x
y
0
0 –3
2
–3
6
-6
y
– 1.5
6
x 2.5
2.5 – 1.5
30
12 3
– 22y 33
y – 1.5
x
y
– 1.5
10
y
x
___________________
3y = 12
Y=4
4x + 4= – 4
4x = – 8
X= –2
Answer: (– 2, 4)
– 2(7x + y = 2)
– 14x – 2y = – 4
– 7x = 0
x=0
5(0)+2y = 4
2y = 4
y=2
Answer: (0,2)
–1
X
–3
–7 3
– 3x
x
x
–3
– 21
x
y
y
2
–3 2
15
9
–3
y
1. Rewrite in standard
form:
x – 3y = 8
3x + 4y = 11
3. Add equations and solve for y:
– 3x + 9y= – 24
3x +4y = 11
0 + 13y = – 13 y = – 1
2. Get the coefficients of x
to be opposite:
– 3( x – 3y = 8)
3x + 4y = 11
4. Substitute – 1 for y:
x – (3)(– 1) – 8 =0
x + 3 – 8=0
x – 5=0
x=5
Answer: (5,-1)
1.Both equations in
standard form:
6x + 5y = 23
9x – 2y = – 32
2. Get the coefficients of
y to be opposite:
2(6x + 5y = 23)
5(9x – 2y= – 32)
Answer: (– 2,7)
3. Add equations an solve
for x:
12x + 10y = 46
45x –10y = – 160
57x + 0 = – 114
x=–2
4.Substitute -2 for x:
6(– 2) + y = 23
– 12+ 5y = 23
5y = 35
y =7
Pounds
Of raisins
Pounds
Of granola
Pounds
Of raisins
x
y
20
4
5
85
x y
20
4x 5y
85
x
4x
y
20
5y
85
substitution
substitute
4
20 – y
80 – 4y
5y
5y
x
5
20 – y
85
85
5
20
15
x
y
20
15
5
20
20
20
5
4x
4
15
5
60
15
5y
85
5
85
25
85
85
85
Pounds of granola + pounds of raisins= total pounds
Price of granola x Pounds of granola +
price of raisins x pounds of raisin = total
1. Write equations
x+ y=30
4x+5y=125
3. Solve for y:
4y+120+5y= 125
y=125-120 ,y = 5lbs raisins
2. Substitute for x :
x= -y+30
4(-y+30) +5y= 125
4. Substitute 5 for y:
x+5=30, x=25 lbs granola
1. Write the equations
3x + 2y = 161,000
2x + 3y= 154,000
2. Get the coefficient x to
be the opposite
– 2(3x + 2y = 161,000)
3(2x + 3y= 154,000)
3. Add and solve for y:
– 6x – 4y = – 322000
6x + 9y = 462000
5y=140,000
y = 28,000
4. Substitute 28,00 for y
3x + 2(28,000) = 161000
3x + 56,000 = 161000
3x = 105,000
x = 35,000
x–3
x+2
Parallel Parallel
intersect
no solution
x+2
x+2
x+2
x+2
2
False
eliminated
false
no solution
– 3x – 1
– 3x – 1
Every
2
6
2
-2
0
0
true
2
True
eliminated
infinitely many solutions
– 2y = – x + 3
Y= 1/2x – 3/2
3y = 2x + 4
Y= 2/3x + 4/3
10y = 5x – 15
Y = 1/2x – 3/2
6y = 4x + 10
Y=2/3x + 5/3
Same equation: Infinitely
many solutions
Same slope so parallel lines:
No Solution
15y = 25x + 2
Y= 5/3x + 2/15
– 3y = – 5x + 7
Y= 5/3x – 7/3
Same slope so parallel
No solution
y=–x–6
– y = – 11x + 42
Y = 11x – 42
11x – 42 = – x – 6
12x = 36
x=3
y= (– 3 – 6)
y= – 9
One Solution: (3, – 9)
x=2
x=–4
x=–4
x<2
x>–4
between
on
x=2
0 1
3
0
2
1
0
0
x>–2
y1
The region that lies on all four half-planes is a quadrilateral
With vertices at (0,0), (4,0), (1, – 3), (0, – 3)