# Question Video: Finding an Augmented Matrix given a System of Equations Mathematics • 10th Grade

Find the augmented matrix for the following system of equations: 8𝑥 − 3𝑧 − 7 = 0, 6𝑦 + 3𝑥 = 0, 7𝑧 − 6𝑦 + 8 = 0.

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### Video Transcript

Find the augmented matrix for the following system of equations: eight 𝑥 minus three 𝑧 minus seven equals zero, six 𝑦 plus three 𝑥 equals zero, and seven 𝑧 minus six 𝑦 plus eight equals zero.

By examining the system of equations given, we can see that this is not written in the most helpful format. In order to minimize our chance of making a mistake when creating the augmented matrix, we begin by rewriting the system of linear equations. We will do this by lining up the 𝑥, 𝑦, 𝑧, and constant terms underneath each other. Adding seven to both sides of our first equation, we have eight 𝑥 minus three 𝑧 is equal to seven. And as there is no 𝑦-term, we will leave a gap here. The second equation has no 𝑧-term. And this can be rewritten as three 𝑥 plus six 𝑦 equals zero. Subtracting eight from both sides of the third equation and then rearranging the left-hand side gives us negative six 𝑦 plus seven 𝑧 is equal to negative eight. We notice that in this equation there is no 𝑥-term.

Having rewritten our equations, we see that the system has three equations in three variables. And hence, the augmented matrix is of dimension three by four. The first column of the augmented matrix contains the coefficients of 𝑥. These are eight, three, and zero. The second column contains the coefficients of 𝑦: zero, six, and negative six. Next, we have the 𝑧-components: negative three, zero, and seven. And finally, we enter the constants on the right-hand side of our equations: seven, zero, and negative eight. The augmented matrix for the system of equations eight 𝑥 minus three 𝑧 minus seven equals zero, six 𝑦 plus three 𝑥 equals zero, and seven 𝑧 minus six 𝑦 plus eight equals zero is eight, zero, negative three, seven, three, six, zero, zero, zero, negative six, seven, negative eight.