Solve the system of equations using the matrix tool and enter the solution as an ordered pair.
3x + 4y = 18
2x + 5y = 19
3x + 4y = 18 .....(1)
2x + 5y = 19 .....(2)
Solving by elimination:
5*(1) - 4*(2) will eliminate y. Solve for x. Substitute the value of x into each equation to see if you get the same values of y.
Solving by substitution:
From (2), divide by 2 and transpose the y-term to get:
x = (19-5y)/2
Substitute x in equation (1) and solve for y. From the value of y obtained substitute into the original equations and solve for x. You should get identical results.
Trying again using matrices.
Let
A =
|3 4|
|2 5|
B =
|18|
|19|
x =
|x|
|y|
then
Ax=B
and
x = A-1B
The inverse of a general 2x2 matrix is given by
A-1
|d -b|
|-c a| divided by |A|
where
A=
|a b|
|c d|
and
|A| is the determinant of A = ad-bc
So in the given problem,
A-1 =
|5 -4|
|-2 3| divided by |A|=5*3-4*2=7
x =
|5 -4| |18|
|-2 3| |19|
=
|18*5 -19*4|
|-2*18+3*19| divided by 7
=
|14/7|
|21/7|
=
|2|
|3|
Check:
|3 4| |2|
|2 5| |3|
=
|18|
|19|
Correct!
An even simpler method for 2x2 matrices is by determinants.
As we have seen above,
|A| = 7
we can calculate x and y directly by calculating the determinant of A after replacing the corresponding column of A by the B vector, and dividing by |A|.
Thus
x=
|18 4|
|19 5| divided by |A|=7
=(5*18-4*19)/7=(90-76)/7=2
y=
|3 18|
|2 19| divided by |A|=7
=(3*19-2*18)/7=(57-36)/7=3
Thus
x=
|2|
|3|
To solve a system of equations using the matrix tool, we can represent the system of equations in the form of a matrix equation AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
First, let's represent the given system of equations in matrix form:
```
| 3 4 | | x | | 18 |
| 2 5 | · | y | = | 19 |
```
We can solve this system by performing matrix operations. Using the formula X = A^(-1)·B, we can find the value of X.
1. Calculate the determinant (det) of matrix A:
```
det(A) = (3 * 5) - (4 * 2) = 15 - 8 = 7
```
2. Find the inverse (A^(-1)) of matrix A:
```
A^(-1) = (1 / det(A)) * adj(A)
```
where adj(A) is the adjugate of matrix A.
To calculate adj(A), we need to find the cofactor matrix of A:
```
| 5 -4 | | 5 -4 |
|-2 3 | -> | -2 3 |
```
Then, transpose the cofactor matrix to get the adjugate of A:
```
adj(A) = | 5 -2 |
|-4 3 |
```
Now, calculate the inverse of A:
```
A^(-1) = (1 / det(A)) * adj(A)
= (1 / 7) * | 5 -2 |
|-4 3 |
= | 5/7 -2/7 |
| -4/7 3/7 |
```
3. Calculate the matrix product X = A^(-1) · B:
```
| 5/7 -2/7 | | 18 | | (5/7)*18 + (-2/7)*19 |
| -4/7 3/7 | · | 19 | = | (-4/7)*18 + (3/7)*19 |
```
Perform the calculations:
```
| 5/7 -2/7 | | 18 | | 90/7 + (-38/7) |
| -4/7 3/7 | · | 19 | = | (-72/7) + (57/7) |
```
Simplify the expression:
```
| 5/7 -2/7 | | 18 | | 52/7 |
| -4/7 3/7 | · | 19 | = | -15/7 |
```
Therefore, the solution to the system of equations is x = 52/7 and y = -15/7.
The solution can be written as an ordered pair (x, y) = (52/7, -15/7).