Here are the solutions to the simultaneous equations using the matrix method:
a) Given the equations:
2x+y=6
x−y=3
Step 1: Write the equations in matrix form AX=B.
[211−1][xy]=[63]
Step 2: Calculate the determinant of matrix A.
det(A)=(2)(−1)−(1)(1)=−2−1=−3
Step 3: Find the inverse of matrix A, A−1.
A−1=det(A)1[−1−1−12]=−31[−1−1−12]=[313131−32]
Step 4: Multiply A−1 by B to find X.
[xy]=[313131−32][63]=[31(6)+31(3)31(6)+(−32)(3)]=[2+12−2]=[30]
Thus, x=3 and y=0.
The solution is x=3,y=0.
b) Given the equations:
5x+3y=3
3y−11=3x
Step 1: Rearrange the second equation to the standard form ax+by=c.
5x+3y=3
−3x+3y=11
Write the equations in matrix form AX=B.
[5−333][xy]=[311]
Step 2: Calculate the determinant of matrix A.
det(A)=(5)(3)−(3)(−3)=15−(−9)=15+9=24
Step 3: Find the inverse of matrix A, A−1.
A−1=det(A)1[33−35]=241[33−35]=[243243−243245]=[8181−81245]
Step 4: Multiply A−1 by B to find X.
[xy]=[8181−81245][311]=[81(3)+(−81)(11)81(3)+245(11)]=[83−81183+2455]=[−88249+2455]=[−12464]=[−138]
Thus, x=−1 and y=38.
The solution is x=−1,y=38.
c) Given the equations:
x+y=3
x−y=3
Step 1: Write the equations in matrix form AX=B.
[111−1][xy]=[33]
Step 2: Calculate the determinant of matrix A.
det(A)=(1)(−1)−(1)(1)=−1−1=−2
Step 3: Find the inverse of matrix A, A−1.
A−1=det(A)1[−1−1−11]=−21[−1−1−11]=[212121−21]
Step 4: Multiply A−1 by B to find X.
[xy]=[212121−21][33]=[21(3)+21(3)21(3)+(−21)(3)]=[23+2323−23]=[260]=[30]
Thus, x=3 and y=0.
The solution is x=3,y=0.
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