# Linear Equations in Two Variables Word Problems

Inverse variation refers to a nonlinear relationship in which the product of two variables is constant. Linear equations are used in many real life fields like engineering, accounting, banking etc. make their jobs easier. Linear equations word problems often intimidate students. Even to those students who are very good at Math find solving word problems difficult. This is usually because word problems require converting the problem into a mathematical equation which can be solved to arrive at an outcome. Even students who are good at Math may not always be good reading a problem and arriving at an equation. Linear equations word problems relate mathematical concepts to real world situations.

## Word Problems

Algebra word problems make use of many an algebraic models to answer questions rising in real life situations. If students solve Math word problems every day, their confidence grows rapidly as they feel more confident about being able to solve them. Usually Math word problems describe real life situations most of the time, so, learning how to solve these problems helps students immensely.
Problem 1:
Aarthi went to US with her Mom for vacations. While returning back to India she gave her savings to her mom to convert into Indian currency. Her mom found that, Aarthi has collected 150 coins in his Piggy Bank, all consisting of dimes and quarters. If the total worth of the coins is 30 dollars, how many dimes and quarters does Aarthi have?

Solution: Let x and y be correspondingly the number of dimes and quarters in the Piggy Bank.
We can form two equations, one on the total number of coins and another on the value of the coins.
x + y = 150  ....(1)
0.10x + 0.25y = 30 ....(2)
Because 1 dime value = 0.10 dollars and 1 quarter value = 0.25 dollars
The system can be solved using substitution as follows:
Solve equation (1) for y
y = 150 - x
Put the value of y in equation (2), to get the value of x
0.10x + 0.25(150 - x) = 30
0.10x + 0.25 $\times$ 150 - 0.25x = 30
0.10x - 0.25x = 30 - 37.5
-0.15x = -7.5
x = 7.5/0.15 = 50

Again from equation (1)
50 + y = 150
y = 150 - 50 = 100
Number of Dimes = 50
Number of Quarters = 100

Problem 2: The linear model P(d) = 62.5d + 2117 is used to find the pressure (lb/ft^2) at d feet below the surface of the water.
(a) What does the constant 2117 represent?
(b) What information do you get from the number 62.5?
(c) What is the pressure 200 ft below water surface?
Solution:
The model is given in slope intercept form y = mx + b , where m is the slope and b the y intercept.
a)The constant 2117, which can be viewed as the y intercept is the function value when d = 0.
This means the pressure on the surface of water = 2117 lb/ft^2.

b)The number 62.5 can be related to 'm' the slope in a linear equation. It is the rate at which the pressure is increasing for every ft below the water surface.
c)
P(d) = 62.5d + 2117
P(200) = 62.5(200) + 2117 = 12,500 + 2,117 = 14,617 lb/ft^2.

Problem 3: The sum of two number is 50 and their difference is 10. Find the numbers.
Solution: Convert the statement into simpler equations:
Consider two numbers, say x and y
Sum of two numbers = x + y = 50
Difference of two numbers = x - y = 10

Add both the equations as coefficient of variable y are same and opposite in signs.
x  + y = 50
x  - y  = 10
_________
2x + 0y = 60

2x  =60
x = 30

Again, put value of x in x - y = 10
30 - y = 10
y = 30 - 10 = 20

Practice Problem 1: If 1 is added to the numerator and denominator of any fraction, it becomes $\frac{1}{2}$. If 2 is subtracted from the numerator and denominator fraction become $\frac{3}{2}$. Find the fractions.