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 xx and yy

Let:

  • xx be the profit from selling one model A computer.
  • yy be the profit from selling one model B computer.

We are given two situations:

  1. The dealer sells 6 model A’s and 9 model B’s, and the total profit is P 585.

    • This gives the equation: 6x+9y=5856x + 9y = 585
  2. The dealer would make the same profit from selling 3 model A’s and 2 model B’s.

    • This gives the equation: 3x+2y=5853x + 2y = 585

Thus, the two simultaneous equations are: 6x + 9y = 585 \tag{1} 3x + 2y = 585 \tag{2}

Part (b) - Solve these equations for xx and yy

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 xx or yy 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): (6x+4y)(6x+9y)=1170585(6x + 4y) - (6x + 9y) = 1170 - 585 6x+4y6x9y=5856x + 4y - 6x - 9y = 585 5y=585-5y = 585 y=117y = -117

Step 4: Solve for xx

Substitute y=117y = -117 into one of the original equations, say equation (3): 2x+3(117)=1952x + 3(-117) = 195 2x351=1952x - 351 = 195 2x=195+3512x = 195 + 351 2x=5462x = 546 x=273x = 273

Final Answer:

The profit per model A computer is x=273x = 273, and the profit per model B computer is y=117y = -117.


Do you have any questions, or would you like further details on any of these steps?

Here are 5 related questions to further explore:

  1. How can we verify the solutions for xx and yy using substitution?
  2. What if the profit equation for model B changes to a different value?
  3. How would the system change if more models were introduced?
  4. Can this method be applied to solve nonlinear systems?
  5. 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