Question

Can somebody give me an review over how to solve systems of equations by substitution and elimination? PLEASE HELP I forgot how to do it and my Algebra EOC is TOMORROW!! I provided an example question and I need LONG ACCURATE and HELPFUL explanations, not a one sentence response (or I will report you). PLEASE HELP ME!!!

Answer 1

Price per large envelope = $1.42 (option A)

Price of each postcard = $0.35 (not asked for in this specific question)

Step-by-step process for solution given below

Explanation:

The first thing to do for this problem and other similar problems is to set up the system of linear equations to be solved

We start with the unknowns and provide variable names for these unknowns

There are two unknowns here - the cost of each postcard and the cost of each large envelope

Let
x = cost of one postcard

y = cost of one large envelope

Customer 1 bought 14 postcards(cost = 14x) and 5 envelops(cost = 5y) for a total cost of $12

We can set up this as an equation:
12x + 5y = 12 ............ [1]

Customer 2 bought 10 postcards(cost = 10x) and 15 envelops(cost = 15y) for a total cost of $24.80
This translates to the following equation:
10x + 15y = 24.80 .... [2]

Let's re-write these equations one below the other for easier solution:

`\begin{amatrix}14x+5y=12\dots[1]\\ 10x+15y=24.8\dots[2]\end{amatrix}`

Solving by substitution:
From any one of the above equations, express one variable in terms of the other variable and substitute for that variable in the other equation to solve.

Take equation [1]

`14x+5y=12`

Let's express x in terms of y.

Move 5y to the right:

`14x=12-5y`

`x =(12-5y)/(14)`

Use this expression in y for x in equation [2]

`10x + 15y = 24.8\\\\\implies 10 \cdot (12-5y)/(14)+15y=24.8\\\\\implies (\left(12-5y\right)\cdot \:10)/(14) + 15y = 24.8\\\\\implies (\left(12-5y\right)\cdot \:10)/(14) + 15y = 24.8\\\\`

Multiply throughout by 14:

`14 \cdot (\left(12-5y\right)\cdot \:10)/(14) + 14\cdot 15y = 14 \cdot 24.8\\\\(12-5y)10 + 210y = 347.2\\\\120 - 50y + 210y = 347.2\\\\120 + 160y = 347.2\\\\160y = 347.2- 120\\\\160y = 227.2\\\\y = (227.2)/(160)\\\\y = 1.42`

Substitute y = 1.42 in equation
`x=(12-5y)/(14)`

`x=(12-5\cdot \:1.42)/(14)\\\\x = 0.35\\\\`

So the cost of each large envelope was $1.42 and cost of each postcard was $0.35

Answer is option A

Solve by elimination

The idea here is to get the coefficients of one of the variables the same in both equations by multiplying / dividing one or both of the equations by some factor and then add/subtract to isolate the other variable and solve

For equations 1 and 2 do the following

• Multiply 14x+5y=12 by 5 : 70x+25y=60
• Multiply 10x+15y=24.8 by 7 : 70x+105y=173.6

Subtract the equations:

`70x+105y=173.6\\-\\70x+25y=\;\;\;\;\;60\\\\`
-------------------------------

`80y=113.6`

y = 113.6/80 = 1.42 as before

Substitute for y in equation 1 and solve for x
14x + 5y = 12 => 14x + 5(1.42) = 12

=> 14x + 7.1 = 12

=> 14x = 12 - 7.1

=> 14x = 4.9

=> x = 4.9/14 = 0.35 as before

I personally prefer solving by elimination since solving by substitution involves heavy duty fractional expressions