Video Transcript
Find the system of linear equations from the following augmented matrix: nine, seven, negative five, four, nine, four, negative seven, five, negative four, negative seven, five, negative nine.
Systems of linear equations can be represented in different ways. In this question, we’ve been given a system of linear equations as an augmented matrix. Augmented matrices look like this, with the coefficients on the left-hand side and the constants written on the right-hand side. We can work out the number of linear equations by the number of rows in the augmented matrix. This matrix has three rows. Therefore, we’re going to have three linear equations.
The left-hand part of the augmented matrix is what we call the coefficient matrix. And because here this has three columns, it means that we have three variables. So let’s say our variables are 𝑥, 𝑦, and 𝑧. Then we can write down the first linear equation by considering the first row of this matrix. Nine, seven, and negative five are the coefficients of the variables for the first linear equation. Therefore, the first linear equation is nine 𝑥 add seven 𝑦 minus five 𝑧. And that is equal to our constant, which is on the right-hand side, and that’s four.
So let’s now use the same method to find our second linear equation. That’s going to be nine 𝑥 add four 𝑦 minus seven 𝑧 equals five. And let’s now do our final linear equation. That’s negative four 𝑥 minus seven 𝑦 plus five 𝑧 equals negative nine. So our answer is option (B) nine 𝑥 plus seven 𝑦 minus five 𝑧 equals four, nine 𝑥 plus four 𝑦 minus seven 𝑧 equals five, and negative four 𝑥 minus seven 𝑦 plus five 𝑧 equals negative nine.
