Math Problem Statement
A computer dealer sells two models of computer A and B. In one particular week he sells 6 model A’s and 9 model B’s. The profit on these sales is P 585. He would make the same profit from selling 3 model of A’s and 2 model B’s. If the profit on a model A computer is P x and on a model B computer is P y,
a) Obtain two simultaneous equations in x and y to express these facts. [4 marks]
b) Solve these equations for x and y. [4 marks]
Solution
Let's break this down step by step:
Part (a) - Obtain two simultaneous equations in and
Let:
- be the profit from selling one model A computer.
- be the profit from selling one model B computer.
We are given two situations:
-
The dealer sells 6 model A’s and 9 model B’s, and the total profit is P 585.
- This gives the equation:
-
The dealer would make the same profit from selling 3 model A’s and 2 model B’s.
- This gives the equation:
Thus, the two simultaneous equations are: 6x + 9y = 585 \tag{1} 3x + 2y = 585 \tag{2}
Part (b) - Solve these equations for and
Now, we solve the system of equations:
Step 1: Simplify Equation (1)
Divide the first equation by 3 to make it easier to work with: 2x + 3y = 195 \tag{3} So now we have two equations: 2x + 3y = 195 \tag{3} 3x + 2y = 585 \tag{2}
Step 2: Eliminate one variable
We can eliminate one variable by multiplying the equations to make the coefficients of or the same.
Multiply equation (3) by 3: 6x + 9y = 585 \tag{4}
Multiply equation (2) by 2: 6x + 4y = 1170 \tag{5}
Step 3: Subtract the equations
Now subtract equation (4) from equation (5):
Step 4: Solve for
Substitute into one of the original equations, say equation (3):
Final Answer:
The profit per model A computer is , and the profit per model B computer is .
Do you have any questions, or would you like further details on any of these steps?
Here are 5 related questions to further explore:
- How can we verify the solutions for and using substitution?
- What if the profit equation for model B changes to a different value?
- How would the system change if more models were introduced?
- Can this method be applied to solve nonlinear systems?
- What impact would it have if negative profits were expected for both models?
Tip: When solving simultaneous equations, it's often useful to simplify one equation before proceeding with elimination or substitution.
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Math Problem Analysis
Mathematical Concepts
Algebra
Simultaneous Equations
Linear Equations
Formulas
6x + 9y = 585
3x + 2y = 585
Theorems
Substitution Method
Elimination Method
Suitable Grade Level
Grades 9-11
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