Using the inverse of:

2 1
-3 4
Find
2x+y=2
-3x+4y=1

It is supposed to be a matrix.

2, 1
-3, 4
but i donot know how to put the brackets. i used commas to separate the numbers

well, I will use gauss jordan, see the link I gave you.

2,1,1,0
-3,4,0,1

+1, .5 , .5, 0
-3, 4. , 0 , 1
multiply top by +3 and add to bottom
+1, .5 , .5 , 0
0 , 5.5, 1.5, 1
divide second row by 5.5
+1, .5 , .5 , 0
0 ,1.0, .273,.182
subtract half the second row from the first to get a new first
+1 , 0 , .364, -.091
0 , 1.0, .273, .182
so I think the inverse is
.364 , -.091
.273 , .182
multiply that by
2
1
and get
x = 2*.364 - 1* .091 = .637
y = 2*.273 +1 * .182 = .728
check
2*.637 + .728 = close enough to 2
-3*.637 + 4*.728 = close enough to 1

I have not inverted a matrix by hand for a while :)

To solve the system of equations using the inverse of a matrix, you need to follow these steps:

Step 1: Write the system of equations in matrix form using the coefficient matrix and the constant matrix.

The coefficient matrix, A, is given by:
A = | 2 1 |
| -3 4 |

The constant matrix, B, is given by:
B = | 2 |
| 1 |

Step 2: Find the inverse of the coefficient matrix, A⁻¹.

To find the inverse of a 2x2 matrix, you can use this formula:
A⁻¹ = 1 / (ad - bc) * | d -b | where a, b, c, and d are the elements of the matrix A.
| -c a |

Using the given values, we have:
A⁻¹ = 1 / ((2 * 4) - (1 * -3)) * | 4 -1 |
| 3/2 1/2 |

Simplifying this, we get:
A⁻¹ = | 4/5 -1/5 |
| 3/10 1/10 |

Step 3: Multiply the inverse of the coefficient matrix, A⁻¹, by the constant matrix, B, to solve for the values of x and y.

Let X be the matrix containing the variables x and y, we have:
X = A⁻¹ * B

Multiplying the matrices, we get:
X = | 4/5 -1/5 | * | 2 |
| 1 |

Simplifying this, we get:
X = | (4/5)*(2) + (-1/5)*(1) |
| (3/10)*(2) + (1/10)*(1) |

Calculating the values, we have:
X = | 8/5 - 1/5 |
| 6/10 + 1/10 |

Simplifying this further, we get:
X = | 7/5 |
| 7/10 |

Therefore, the solution to the system of equations is x = 7/5 and y = 7/10.