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 Depdendent Variable

 Number of equations to solve: 23456789
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 Solve for:

 Dependent Variable

 Number of inequalities to solve: 23456789
 Ineq. #1:
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 Solve for:

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# Solutions to Linear Equations in Two Variables

1. Given 2x − 3y = 6 build the table for using an arithmetic sequence as replacement values.

Logically, we know that it is easy to plot ordered pairs if both coordinates are integers.

In this equation we note that all of the numbers except the y-coefficient are even integers.

From arithmetic we know that:

 Multiplying by any even integer will yield an even integer as the product and Adding two even integers will result in an even integer as the sum. Even Ã— Odd (or Even) = Even   Even Â± Even = Even

Now if we choose an even integer as the replacement for the y- value then the resulting x-value will be an integer. Therefore, we put the "sequence of even integers" in the table for y then complete the computation below for each y-value and put each corresponding x-value in the table. Repeat the process for the other y-values to find the resulting x-values. Note any sequences and write the common differences.

 2x − 3y = 6 Let y = - 4: 2x − 3(- 4) = 6, repeat for the other numbers y = - 2: 2x − 3(- 2) = 6,  y = 0: 2x − 3( 0 ) = 6,  y = 2: 2x − 3( 2 ) = 6,  y = 4: 2x − 3( 4 ) = 6, x = - 3   x = 0 x = 3 x = 6 x = 9 Check to see that both columns of values are arithmetic sequences.

For the point (- 3, - 4) replace x = - 3 and y = - 4 in the given equation 2x − 3y = 6

CHECK:

For the point (9, 4) replace x = 9 and y = 4 in the given equation 2x − 3y = 6

CHECK:

You should always form the practice of checking all of your work whenever you can.

2. Graph 2x + 3y = 7. Build a table for 2x + 3y = 7.

Again let us turn to LOGIC: We know that for integers:

 Odd Ã— Odd = Odd Odd Â± Odd = Even Even Ã— Odd (or Even) = Even Even Â± Odd = Odd

In order to keep both coordinates as integers choose the replacement values for the y to be odd integers. We choose the sequence of odd integers for replacement for y- values and the resulting x-sequence will be integers. Complete the computation below and put the number in the table. Repeat the process for the other y-values to find the resulting x-values.

 2x + 3y = 7      Let y = 1  repeat for the other numbers Put these values in the table, then write the â€œcommon differencesâ€ for both x and y. Use the value for dx to complete the table values for x. For complete table see below. Note adding Â± 3 to given values forms an arithmetic sequence. To be sure you should always check the â€œouter pointsâ€.

Check:

Check:

Check the dy and dx on your table.