{"id":45887,"date":"2025-07-01T13:46:16","date_gmt":"2025-07-01T13:46:16","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=45887"},"modified":"2025-07-01T13:46:18","modified_gmt":"2025-07-01T13:46:18","slug":"solve-the-system-of-linear-equations-using-cramers","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/solve-the-system-of-linear-equations-using-cramers\/","title":{"rendered":"Solve the system of linear equations using Cramer’s"},"content":{"rendered":"\n
Solve the system of linear equations using Cramer's<\/pre>\n\n\n\n
\"\"<\/figure>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n
Here are the solutions to the problem:<\/p>\n\n\n\n
D = 92<\/strong>
Dx = -276<\/strong>
Dy = 276<\/strong>
Dz = 92<\/strong>
The solution is: (x, y, z) = (-3, 3, 1)<\/strong><\/p>\n\n\n\n
Explanation<\/h3>\n\n\n\n
This problem requires solving a system of three linear equations using Cramer’s rule. The given system is:
x + 4y + 4z = 13
-x – y + 3z = 3
6x – 2y – 6z = -30<\/p>\n\n\n\n
First, we must find the determinant of the coefficient matrix, denoted as D. The coefficient matrix is formed using the coefficients of the x, y, and z variables.<\/p>\n\n\n\n
The coefficient matrix (A) is:
| 1 4 4 |
| -1 -1 3 |
| 6 -2 -6 |<\/p>\n\n\n\n
The determinant D is calculated by expanding along the first row:
D = 1 * ((-1)(-6) – (3)(-2)) – 4 * ((-1)(-6) – (3)(6)) + 4 * ((-1)(-2) – (-1)(6))
D = 1 * (6 + 6) – 4 * (6 – 18) + 4 * (2 + 6)
D = 1 * (12) – 4 * (-12) + 4 * (8)
D = 12 + 48 + 32 = 92.<\/p>\n\n\n\n
Next, we find the determinant Dx. We form a new matrix by replacing the first column (x coefficients) of the coefficient matrix with the constant terms from the equations (13, 3, -30).
The matrix for Dx is:
| 13 4 4 |
| 3 -1 3 |
| -30 -2 -6 |
Dx = 13 * ((-1)(-6) – (3)(-2)) – 4 * ((3)(-6) – (3)(-30)) + 4 * ((3)(-2) – (-1)(-30))
Dx = 13 * (12) – 4 * (72) + 4 * (-36)
Dx = 156 – 288 – 144 = -276.<\/p>\n\n\n\n
Then, we find the determinant Dy by replacing the second column (y coefficients) with the constant terms.
The matrix for Dy is:
| 1 13 4 |
| -1 3 3 |
| 6 -30 -6 |
Dy = 1 * ((3)(-6) – (3)(-30)) – 13 * ((-1)(-6) – (3)(6)) + 4 * ((-1)(-30) – (3)(6))
Dy = 1 * (72) – 13 * (-12) + 4 * (12)
Dy = 72 + 156 + 48 = 276.<\/p>\n\n\n\n
After that, we find the determinant Dz by replacing the third column (z coefficients) with the constant terms.
The matrix for Dz is:
| 1 4 13 |
| -1 -1 3 |
| 6 -2 -30 |
Dz = 1 * ((-1)(-30) – (3)(-2)) – 4 * ((-1)(-30) – (3)(6)) + 13 * ((-1)(-2) – (-1)(6))
Dz = 1 * (36) – 4 * (12) + 13 * (8)
Dz = 36 – 48 + 104 = 92.<\/p>\n\n\n\n
Finally, we use Cramer’s rule to find the values of x, y, and z.
x = Dx \/ D = -276 \/ 92 = -3
y = Dy \/ D = 276 \/ 92 = 3
z = Dz \/ D = 92 \/ 92 = 1<\/p>\n\n\n\n
Therefore, the unique solution to the system of equations is (x, y, z) = (-3, 3, 1).<\/p>\n\n\n\n
\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"
Solve the system of linear equations using Cramer’s The Correct Answer and Explanation is: Here are the solutions to the problem: D = 92Dx = -276Dy = 276Dz = 92The solution is: (x, y, z) = (-3, 3, 1) Explanation This problem requires solving a system of three linear equations using Cramer’s rule. The given […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-45887","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/45887","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=45887"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/45887\/revisions"}],"predecessor-version":[{"id":45890,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/45887\/revisions\/45890"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=45887"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=45887"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=45887"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}