SIMULTANEOUS LINEAR EQUATIONS / SYSTEM OF LINEAR EQUATIONS
• An equation is a statement that two expressions are equal. The two expressions that make up an equation are called its sides or members. They are separated by the equality sign, ‘=”. • A linear equation in the variable x is an equation that can be written in the form ax + b = 0, where a and b are constants and a ≠ 0.
• A linear equation contains unknowns only to the first degree, for example 2x + y = 6.
• Simultaneous equations are sets of equations with the same unknown values.
• The main methods of solution are substitution or elimination by adding and subtracting.
1. One-variable system
a. x + 2 = 3
x = 3 - 2
x = 1
b. 5x – 6 = 3x
5x – 3x = 6
2x = 6
x = 3
c. 2 (p + 4) = 7p + 2
2p + 8 = 7p + 2
8 – 2 = 7p – 2p
6 = 5p
6/5 = p
2. Two-variable system
Substitution method
a) Direct substitution
y = 2x - 3 ----------------- (i)
5y + 6x = 1 ----------------- (ii)
Substitute y = 2x - 3
6x + 5 (2x - 3) = 1
6x + 10x - 15 = 1
16x = 16
x = 1
Substitute x = 1 in equation y = 2x - 3
y = 2(1) - 3
y= -1
b) Alteration the equation
x + 4y = 5 -------------- (i)
3y +43x = -6 --------------- (ii)
Substitute x = 5 – 4y in equation (ii)
4(5-4y) + 3y = 6
20 – 16y + 3y = -6
-13y = -26
y = 2
Put y = 2 in equation (i)
x + 4y = 5
x + 4(2) = 5
x = 5 - 8
x = -3
Elimination / Cancellation Method
a. 2x – y = 5 ----------------- (i)
4x + y = 13 ----------------- (ii)
Equation 1 + Equation 2
6x = 18
x = 3
Substitute x = 3 in equation (i)
2x – y = 5
2(3) – y = 5
y = 1
b. Multiply 1 equation with the number
4x - y = 8 -------------- (i)
3x + 2y = 17 --------------- (ii)
Equation 1 X2
8x – 2y = 16 ------------- (iii)
3x + 2y = 17 ------------- (iv)
Equation 3 + Equation 2
11x = 33
x = 3
Subtitute x = 3 in equation 1
4x – y = 8
4(3) – y = 8
y = 4
c. Multiply both number of equation
5x + 3y = -1 -------------- (i)
2x + 7y = 7 --------------- (ii)
Equation 1 X2 and equation 2 X5
10x + 6y = -2 ------------- (iii)
10x + 35y = 35 ------------ (iv)
Equation 3 - Equation 4
-29y = -37
y = 37 / 29
Put y = 37 / 29 in equation 1
5x + 3y = -1
5x + 3(37 / 29) = -1
5x + 3.83 = -1
5x = -1 – 3.83
x = -0.97
3. Three-variable system
Example
x + y + z = 0 --------------- (i)
4x + 2y + z = 2 ----------------(ii)
9x + 3y + z = 10 ------------- (iii)
Equation (ii) - equation (i)
4x + 2y + z = 2
(-) x + y + z = 0
3x + y = 2 -------------- (iv)
Equation (ii) – equation (iii)
4x + 2y + z = 2
(-) 9x + 3y + z = 10
-5y - y = -8
Equation (iv) – equation (v)
3x + y = 2
5x - y = -8
-2x = -6
x = 3
Substitute x = 3 in equation (iv)
3x + y = 2
3 (3) + y = 2
y = 2 -9
y = -7
Substitute x = 3 and y = -7 in equation (i)
x + y + z = 0
x + (-7) + z = 0
-4 + z = 0
z = 4
x = 3 , y = -7 , z = 4
Exercises
Question 1
x + 2y + z = 3 --------------- (i)
x - 3y - z = 2 ----------------(ii)
x - 2y - 2z = -2 ------------- (iii)
Question 2
x - 5y + 3z = 9 --------------- (i)
2x - y + 4z = 6 ----------------(ii)
3x - 2y + z = 2 ------------- (iii)
Answers
1. x = 2, , y = -1, z = 3
2. x = -1 / 2 , y = -1 , z = 3 / 2
CONSUMPTION FUNCTION
A consumption function emphasizes the relationship between total household consumption in an economy and the level of national income. A more general demand for consumption goods might also focus on other factors like the price of goods. The consumption function is a mathematical formula laid out by famed economist John Maynard Keynes. The formula was designed to show the relationship between real disposable income and consumer spending, the latter variable being what Keynes considered the most important determinant of short-term demand in an economy.
The Keynesian Consumption Function
Disposable Income (Yd) = Gross Income - (Deductions from Direct Taxation + Benefits)
The standard Keynesian consumption function is as follows:
C = a + b Yd where,
C= Consumer expenditure
a = autonomous consumption. This is the level of consumption that would take place even if income was zero. If an individual's income fell to zero some of his existing spending could be sustained by using savings. This is known as dis-saving.
b = consumption influenced by national income
Yd = disposable income
There is a positive relationship between disposable income (Yd) and consumer spending (Ct). The gradient of the consumption curve gives the marginal propensity to consume. As income rises, so does total consumer demand.
A change in the marginal propensity to consume causes a pivotal change in the consumption function. In this case the marginal propensity to consume has fallen leading to a fall in consumption at each level of income.
Example
The following table shows the level of household consumption at various levels of disposable income.
Disposable income (RM million) Consumption expenditure (RM million)
0 500
1000 1000
2000 1500
3000 2000
Solution
The consumption function can be written as:
C = a + bY
a = 500
b = ΔC/ ΔY
= 1500 – 1000/(2000 – 1000)
= 0.5
Therefore,
C = a + bY
C = 500 + 0.5Y
Question 1
In a closed economy without government intervention, the consumption function is C = 100 + 0.8Y.
Complete the following table
National Incone(Y)(RM) Consumption( C )(RM)
0
1000
1500
2000
3000
4000
Question 2
In a closed economy without government intervention, the consumption function is C = 200 + 0.8Y.
Complete the following table
National Incone(Y)(RM) Consumption( C )(RM)
0
1000
1500
2000
2500
3000
Answers
Question 1
0, 900, 1300, 1700, 2500, 3300
Question 2
200, 1000, 1400, 1800, 2600, 3400
Factors that Influence Consumption
1. Disposable income or national income
Consumption has a direct relationship with disposable income or national income. A high disposable income will result in high consumption and vice versa.
2. Marginal propensity to consume
Marginal propensity to consume (MPC) is the ratio between change in consumption and change in disposable income. A high MPC will result in a high consumption and vice versa.
3. Income distribution pattern
a. Uneven distribution of income in an economy will effect consumption. Poor citizens will have a higher marginal propensity to consume (MPC) compared to richer citizens
b. Therefore, poorer citizens will utilize a large portion of their income and any increase in income on consumption expenditure. However, for richer citizens, a large portion of their income and any additional income will be saved.
4. Wealth and property
a. Wealth owned by individuals can also influence the level of consumption. Income generated from ownership of property such as houses or land will increase individual ability to incur consumption expenditure.
b. Therefore, an individual who owns a lot of property will have high individual purchasing power to buy goods and services and high household consumption expenditure.
5. Credit and payment installment facilities
a. Credit and payment installment facilities offered by firms and financial institutions will encourage household consumption. Consumers can purchase durable goods like houses, cars, radios, and televisions through payment installment schemes.
b. Credit facilities offered by banks and financial institutions encourage individuals to borrow money. Consumption through loans can be encouraged by reducing the down payment and extending the loan repayment period.
6. Consumer predictions about future price charges
If consumers expect the price of a good to increase in the future, they may increase their present consumption of the god. However, if consumers expect the price of a good to decline in the future, they will defer purchase of the good.
7. Invention of new products
The creation of new products in the market will also influence the level of consumption. Household consumption will increase when new goods are available in the market.
8. Current economic conditions
Individual economic expenditure is influenced by current economic conditions. When national economy experiences deflation, unemployment will occur. Therefore, individual consumption expenditure will decrease. However, when the economy experiences inflation, individuals tend to increase consumption expenditure. Therefore, the consumption of goods and services will increase.
9. Interest rates
Interest rates have a negative relationship with consumption functions. When interest rates are low, households will increase expenditure and decrease expenditure and increase savings to take advantage of higher interest rates.
MARKET EQUILIBRIUM
Def:
• Market equilibrium is a situation in which there is no tendency for price or quantity to change. Therefore, in a condition of equilibrium, price and quantity are stable.
• A market is in equilibrium when the market demand curve and the market supply curve intersect.
• Market equilibrium occurs when the demand curve and the supply curve intersect
Qd = Qs
where
Qd = Quantity demanded
Qs = Quantity supplied
• A market will not be in equilibrium if the quantity demanded is not equal to the quantity supplied. This may happen in two circumstances:
a) Excess supply (surplus)
b) Excess demand (shortage)
Market Equilibrium can be determine by:
i. Numerical Analysis
ii. Graphical Analysis
iii. Mathematical Analysis
Numerical Analysis
Price (RM) Qd Qs
50 200 1000 Excess Supply / Surplus
40 400 800 Excess Supply / Surplus
30 600 600 Equilibrium
20 800 400 Excess demand
10 1000 200 Excess demand
• At the price RM30 per unit, the quantity demanded is the same as the quantity supplied, i.e 600 units. Therefore, the equilibrium price is RM30 and the equilibrium quantity is 600 units
Graphical Analysis
• At the price RM30 per unit, the quantity demanded is the same as the quantity supplied, i.e 600 units. Therefore, the equilibrium price is RM30 and the equilibrium quantity is 600 units
Mathematical Analysis
Qd = a – bP DD equation
Qs = c + dP SS equation
a = the quantity of the demanded good when the price (P) is zero. This value will intersect
the quantity axis
b = the gradient of the demand curve
P = price
c = quantity of goods supplied when the price (P) is zero. This value intersects the quantity axis
b = the gradient of the supply curve
Qd = Qs
a - bP = c + dP
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