Question

1st Question

3x + 8y = -4

2x - 4y = 16

2nd Question

6x - y = 16

x = 4y - 5

3rd Question

x + y = 19

3x - 2y = -3

Answer 1

9514 1404 393

Answer:

1. (x, y) = (4, -2)
2. (x, y) = (3, 2)
3. (x, y) = (7, 12)

Explanation:

When you have a lot of the same kind of problem to solve, it makes a certain amount of sense to let a spreadsheet do it. You can enter the formulas once, and then all you need to do is supply the different data for the different problems.

Here, we can put the equations into standard form, then use Cramer's Rule to solve them. For the two equations ...

• ax +by = c
• dx +ey = g

we can define ...

D = (ae -db)

Then the values of x and y are ...

x = (ce -gb)/D

y = (ag -dc)/D

The attached spreadsheet implements these equations to find the values of x and y.

__

The second equation of the 2nd Question is not given in standard form. To put it into standard form, we need to subtract 4y from both sides:

x -4y = -5

_____

Additional comment

Many graphing calculators have matrix functions that will solve sets of linear equations, too. For those, you can use the function that puts the coefficient matrix into Reduced Row-Echelon Form. Often that function is named RREF( ), as it is on my TI-84 and HP-48 calculators. The input coefficient matrix will look like ...

`\left[\begin{array}{ccc}a&b&c\\d&e&g\end{array}\right]`

and the output matrix will look like ...

`\left[\begin{array}{ccc}1&0&\text{x-value}\\0&1&\text{y-value}\end{array}\right]`

Effectively, the equations are solved by "elimination." If the calculator shows [0 0 0] in the bottom row of output, it means the equations are dependent and there are infinite solutions. If it shows [0 0 1], the equations are inconsistent, and there are no solutions.