SIMULTANEOUS EQUATIONS INVOLVING ONE LINEAR AND ONE QUADRATIC
One of the equations is in linear form while the other is in quadratic form.
Note: One linear, one quadratic is only possible analytically using substitution method.
Examples:
1. Solve simultaneously for x and y (i.e. the points of their intersection)
3x + y = 10 & 2x2 +y2 = 19
Solution
3x + y = 10 ———– eq 1
2x2 + y2 = 19 ——— eq 2
Make y the subject in eq 1 (linear equation)
y = 10 – 3x ———- eq 3
Substitute eq 3 into eq 2
2x2 + (10-3x) 2 = 19
2x2+ (10 – 3x) (10 – 3x) = 19
2x2 + 100 – 30x – 30x + 9x2 = 19
2x2 + 9x2 – 30x – 30x + 100 – 19 = 0
11x2 – 60x + 81 = 0
11x2 – 33x – 27x + 81= 0
11x (x-3) – 27 (x – 3) = 0
(11x – 27) (x – 3) = 0
11x – 27 = 0 or x-3 = 0
11x = 27 or x = 3
\ x = 27/11 or 3
Substitute the values of x into eq 3.
When x = 3
y = 10 – 3(x)
y = 10 – 3(3)
y = 10 – 9 = 1
When x =27/11
y = 10 – 3(27/11)
y = 10 – 51/11
y = (110 – 51)/11
y = 59/11
\when x = 3, y = 1
x = 27/11 , y = 59/11
MORE EXAMPLES
Solve simultaneously for x and y.
3x – y = 3 ——– eq 1
9x2 – y 2 = 45 ——— eq 2
Solution
From eq 2
(3x)2 – y 2 = 45
(3x-y) (3x+y) = 45 ———- eq 3
Substitute eq 1 into eq 3
3 (3x + y) = 45
3x + y = 15 ……………..eq4
Solve eq 1 and eq 4 simultaneously.
3x – y = 3 ——— eq 1
3x + y = 15 ——– eq 4
eq 1 + eq 4
6x = 18
x = 18/ 6
x = 3
Substitute x = 3 into eq 4.
3x + y = 15
3 (3) + y = 15
9 + y = 15
y = 15 – 9
y = 6
\ x = 3, y = 6
Evaluation
Solve for x and y in the following pairs of equations
1. (a) 4x2 – y2 = 15 (b) 3x2 +5xy –y2 =3
2x – y = 5 x – y = 4
WORD PROBLEMS LEADING TO LINEAR AND QUADRATIC EQUATIONS
Example
The product of two numbers is 12. The sum of the larger number and twice the smaller number is 11. Find the two numbers.
Solution
Let x = the larger number
y = the smaller number
Product, x y = 12 …………….eq1
From the last statement,
x + 2y = 11 ………….. eq2
From eq2, x = 11 – 2y ……………eq3
Sub. Into eq1
y(11 – 2y) = 12
11y – 2y2 = 12
2y2 -11y + 12 = 0
2y2 – 8y – 3y + 12 = 0
2y(y-4) – 3(y-4) = 0
(2y-3)(y-4) =0
2y-3 =0 or y-4 =0
2y = 3 or y = 4
y= 3/2 or 4
when y = 3/2 when y=4
x = 11 – 2y x = 11- 2y
x = 11 – 2(3/2) x = 11 – 2(4)
x = 11 – 3 x = 11 – 8
x = 8 x = 3
Therefore, (8 , 3/2)(3 , 4)
Evaluation
Solve the following simultaneous equation
1. (a) 22x-3y = 32, 3x-2y = 81 (b) 2x+2y=1, 32x+y = 27
2. Bisi’s and Fibie’s ages add up to 29. Seven years ago Bisi was twice as old as Fibie. Find their present ages.
GENERAL EVALUATION AND REVISION QUESTIONS
1. Solve the simultaneous equation: 3x2 – 4y = -1 & 2x – y = 1
2. Five years ago, a father was 3 times as old as his son, now their combined ages amount to 110 years. How old are they?
3. Solve: 4x2 – y2 = 15 & 2x – y = 5
4. Seven cups and eight plates cost # 1750. Eight cups and seven plates cost #1700. Calculate the cost of a cup and of a plate.
WEEKEND ASSIGNMENT
Solve each of the following pairs of equations simultaneously,
1. xy = -12 ; x – y = 7 a. (3 , -4)(4 ,-3) b. (-2 ,4)(-3, -4) c.(-4, 5)(-2 , 3) d.(3 ,-3)(4,-4)
2. x – 5y = 5 ; x2 – 25y2 = 55 a (-8, 0)(3/5 , 0) b. (0, 0)(-8 , 3/5) c. (8 , 3/5) d. (0, 8)(0, 3/5)
3. y = x2 and y = x + 6 (a).(0,6) (3,9) (b)(-3,0) (2,4) (c) (-2,4) (3,9) (d).(-2, 3), (-3,2)
4. x – y = -3/2 ; 4x2 + 2xy – y2 = 11/4 : a. (-1, 1/2)(1, 5/2). b. (3, 2/5) (1, 1/2) c.(3/2 , -1) (4,2) d.(-1 , -1/2)(-1 , 5/2)
5. m2 + n2 = 25 ; 2m + n – 5 = 0 : a. (0,5)(4, -3) b.(5,0)(-3,4)c.(4,0)(-3,5) d(-5,3)(0,4)
THEORY
1a. Find the coordinate of the points where the line 2x – y = 5 meets the curve 3x2 – xy -4 =10
b. Solve the simultaneous equation: 22x+4y = 4, 33x + 5y – 81= 0
2. A woman is q years old while her son is p years old. The sum of their ages is equal to twice the difference of their ages. The product of their ages is 675.
Write down the equations connecting their ages and solve the equations in order to find the ages of the woman and her son. (WAEC)
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