# Tag Archives: system of equations

## Real Solutions (x,y)

How many real solutions are there that satisfy the two equations and ? Source: NCTM Mathematics Teacher, September 2006 Solution The graph of is a circle centered at the origin and radius = . The graph of is a hyperbola … Continue reading

## How Old Are They?

Mary’s and Bob’s ages combined are twice Jane’s age. Mary is years older than Bob. Jane’s age plus Bob’s age is years. How old are they? Source: NCTM Mathematics Teacher, September 2006 Solution Let , and represent the ages of … Continue reading

## Values of Integers

If and are integers such that , what are the possible values of ? (a) only (b) only (c) only multiples of (d) any integer (e) none of these Source: NCTM Mathematics Teacher, September 2006 Solution Let be an integer … Continue reading

## Lemonade Stand

John has a lemonade stand. He sells a small lemonade for cents and a large lemonade for . A small serving contains cup of lemonade; a large contains cups. At the end of the day, John has made and sold … Continue reading

## Triples of Real Numbers

What are all ordered triples of real numbers that satisfy ? Source: NCTM Mathematics Teacher, October 2006 Solution Add the three equations Substitute the value of into the equations Solve for yields . Substitute the value of into the equations Solve … Continue reading

## Largest of Five Integers

A set contains five integers. When distinct elements of this set are added together, two at a time, the complete list of different possible sums is , and . What is the largest of the five integers in the set? … Continue reading

## Absolute Value

How many different real-valued pairs satisfy this system of two equations? Source: NCTM Mathematics Teacher, November 2006 SOLUTION Suppose Substitute the value of into Eq. or We have two pairs and . Suppose Substitute the value of into Eq. or … Continue reading

## Value of a+b+c

There are positive integers , and that satisfy this system of two equations: What is the value of ? Source: NCTM Mathematics Teacher, November 2006 SOLUTION Multiply the first equation by , the second equation by and add them because is a … Continue reading

## Function Value

Suppose that is a function such that for every real number . What is the value of ? Source: NCTM Mathematics Teacher, November 2006 SOLUTION When When Multiply Eq. by and Eq. by and add them Answer:

## No Integer Solutions

1. Show that the following system of equations has no integer solutions 2. Show that the following system of equations has no integer solutions Source: http://www.math.rutgers.edu/~erowland/modulararithmetic.html SOLUTION 1. We reduce the equations and hope to find a contradiction thereby showing … Continue reading