What is the difference between an equation with two variables and an equation with three variables?
one has two, as x and y
one has three, as x, y and z
As bobpursley said one would have variables such as "x" and "y" and the other would have variables of "x", "y' and "z". The one with variables of x and y could be solved for either x or y and substiruted into the one with variables of x, y and z yielding you a new equation with variables of either x and z or y and z which can now be solved by the method of successive reductions or the Euclidian Algorithm.
Example of successive reductions:
Janet has $8.55 in nickels, dimes, and quarters. She has 7 more dimes than
>nickels and quarters combined. How many of each coin does she have?
>
1--.05N + .10D + .25Q = 8.55
2--5N + 10D + 25Q = 855
3--D = N + Q + 7
4--Substituting, 5N + 10N + 10Q + 70 + 25Q = 855.
5--Collecting terms, 15N + 35Q = 785 or 3N + 7Q = 157, an equation with 2 variables.
6--Dividing through by the lowest coefficiet, 3 yields N + 2Q + Q/3 = 52 + 1
7--(Q - 1)/3 must be an integer k making Q = 3k + 1
8--Substituting back into (5) yields 3N + 21k + 7 = 157 or N = 50 - 7k
9--k can be any value from 0 through 7
10--k....0....1....2....3....4....5....6....7
.....N...50...43..36..29..22..15...8....1
.....Q....1....4....7...10..13..16..19..22
.....D...58...54..50..46..42..38..34..30
11--Therefore, there are 8 solutions.
Example of Euclidian Algorithm:
What is the smallest positive integer that leaves a remainder of 1 when divided by 1000 and a remainder of 8 when divided by 761?
This can be expressed by 1000x + 1 = 761y + 8 = N.
Rearranging, 1000x - 761y = 7, an equation of 2 variables.
First find a solution to 1000x - 761y = 1
Using the Euclidian Algorithm:
1000 = 1(761) + 239
761 = 3(239) + 44
239 = 5(44) + 19
44 = 2(19) + 6
19 = 3(6) + 1
Then
1 = 19 - 3(6)
1 = 19 - 3(440 + 6(19) = 7(19) - 3(44)
1 = 7(239) - 35(440 - 3(44) = 7(239) - 38(44)
1 = 7(239) - 38(761) + 114(239) = 121(230) - 38(761)
1 = 121(1000) - 121(761) - 38(761) = 121(1000) - 159(761)1 =
Therefore, x = 121 and y = 159 is one solution to 1000x - 761y = 1
Multiplying by 7 yields x = 847 and y = 1113, as a solution to 1000x - 761y = 7.
The general solution is then x = 847 - 761t and y = 1113 - 1000t.
The smallest solution occurs when t = 1 yielding x = 86 and y = 113.
This then permits the definition of the positive solutions as x = 86 + 761t and y = 113 + 1000t.
8 divided by 761
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An equation with two variables and an equation with three variables differ in the number of unknowns or variables involved.
To understand the difference between the two, let's start with an equation with two variables. Such an equation typically represents a relationship between two quantities. We can denote the variables as x and y. For example, the equation y = 2x represents a linear relationship where every y value is twice the corresponding x value.
When working with an equation with two variables, the goal is often to find specific values for x and y that satisfy the equation. This could involve solving for one variable in terms of the other, finding specific points on a graph, or using other methods like substitution or elimination.
On the other hand, an equation with three variables involves a relationship between three quantities. Let's denote the variables as x, y, and z. For example, the equation 2x + 3y - z = 10 represents a linear relationship between three variables.
Solving an equation with three variables typically requires finding specific values for all three variables that satisfy the equation. This process involves more steps and techniques compared to equations with two variables. Methods like substitution, elimination, or solving equations simultaneously might be necessary to determine the values of x, y, and z.
In summary, the main difference between equations with two variables and equations with three variables lies in the number of unknowns or variables involved, which affects the complexity of the problem-solving process.