Systems of Simultaneous Linear Equations: Numerical Problems

 What Are Systems of Simultaneous Linear Equations?

A system of simultaneous linear equations is a collection of two or more linear equations involving multiple variables, where the equations are solved together because the variables are interdependent.

Each equation represents a relationship between variables, and the solution is a set of values that satisfies all equations at the same time.

Its uses in Economics

In economics, many variables are interrelated. For instance:

• Consumption depends on income,
• Income depends on investment,
• Investment may depend on interest rates.

These relationships are modeled using simultaneous equations to understand how changes in one variable affect others within an economic system.

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NP1: Solve the given system of  simultaneous equations:

Solution:

From (2):

Substitute into (1):

Now plug y=2y = 2y=2 into (2):

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NP2: Solve the given system of  simultaneous equations:

Solution:

Multiply (1) by 2:

 

Subtract (2) from (3):

Now plug into (1):

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Application in Economics:

NP1: A commodity is produced by 3 units of labour and 2 units of capital. The total cost comes to 62. If the commodity is produced by using 4 units of labour and 1 unit of capital, the cost comes to 56. What is the cost per unit of labour and capital?

Solution:

Given:

Where:

• L is the cost per unit of labour
• K is the cost per unit of capital

Solving by elimination method:

To do that, multiply Equation 2 by 2:

Subtract (1) from (3):

Substitute  in (2)

Cost per unit of Labour (L) = ₹10 and cost per unit of Capital (K) = ₹16

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NP2: A product is made using 5 units of labour and 2 units of capital, costing ₹74. Another product uses 3 units of labour and 4 units of capital, costing ₹78. Find the cost per unit of labour and capital.

Solution:

Given:

Where:

L is the cost per unit of labour

K is the cost per unit of capital

Solving by elimination method:

To do that, multiply Equation (1) by 2:

Subtract (2) from (3):

Substitute  in (1)

⇒

Cost per unit of Labour (L) = ₹10 and cost per unit of Capital (K) = ₹12

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NP3: A consumer buys two goods: Good A and Good B. When they buy 4 units of A and 6 units of B, they spend ₹72. When they buy 6 units of A and 3 units of B, they spend ₹75. Find the price per unit of Good A and Good B.

Solution:

Given:

Let:

A = price per unit of Good A

B = price per unit of Good B

Solving by elimination method:

To do that, multiply Equation (2) by 2:

Subtract (2) from (3):

Substitute  in (1)

⇒

⇒

⇒

⇒

The price per unit of Good A = 9.75 and Good B = 5.5

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NP4: A producer uses 2 tonnes of steel and 3 tonnes of copper to make a product costing ₹132. Another variation uses 5 tonnes of steel and 1 tonne of copper, costing ₹140. Find price per tonne of steel and copper.

Solution:

Given:

Let:

S = price of steel

C = price of copper

Solving by elimination method:

To do that, multiply Equation (2) by 3:

Subtract (1) from (3):

Substitute  in (1)

⇒

⇒

⇒
The price per tonne of steel and copper are:

Steel = ₹22.15/tonne

Copper = ₹29.25/tonne

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