WEBVTT
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Use determinants to solve the system.
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Given a system of linear equations, Cramer’s rule is a handy way to solve for just one of the variables without having to solve the whole system of equations.
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However, in this case they want us to find all of them: 𝑥, 𝑦, and 𝑧.
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So let’s first begin by taking our constants and moving them to the right-hand side.
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This way we can make this in terms of a matrix equation.
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Now that the constants are isolated, over on the right-hand side of the equation, we can turn this into a matrix equation.
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So first let’s take all of the coefficients and put it into a matrix.
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Next, we need to take this matrix and multiply it by the 𝑥𝑦𝑧 matrix.
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Our answer column which is one, negative three, negative one.
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Cramer’s rule states that we can solve for 𝑥, 𝑦, and 𝑧 using this formula.
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So what are all these symbols meaning?
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The triangle itself is the matrix of coefficients which is located here.
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So this triangle 𝑥 means it’s the matrix where the 𝑥 column will be replaced with the answer column, same thing with the 𝑦 and the 𝑧.
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So lastly, what are all of these lines that look like absolute value lines.
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Those represent, to take the determinant.
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So we will need to take the determinate of each of these matrices.
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So let’s begin with finding the determinant of the coefficient matrix, which is located on the denominator of every single fraction.
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So the determinant of this big three-by-three matrix will begin by taking the top left-hand corner number, three.
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And multiplying by, and we multiply by the determinant of the numbers that are not in the row or column of the three that we began with.
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And then we subtract the top middle number, two, times the determinant of all of the numbers that are not in the row or column of the two.
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And then we add negative two times the determinant of these numbers, the numbers not- that are not in the row or column of the negative two.
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So we take three times this determinant.
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So how do we find a determinant?
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We multiply the numbers that are diagonal and then subtract the other numbers that are diagonal, and now we repeat.
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So now we need to subtract two times, begin with the top left-hand corner number, three, and negative five, we multiply and then we subtract.
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And after subtracting negative two times negative three, now we take negative two and multiply by this determinant, three times four minus negative two times three.
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And now we simplify.
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So we have three times negative three minus two times negative 21 minus two times 18, and we get negative three which is the answer to the determinant of the matrix coefficient.
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So we can replace all of the dominators with negative three.
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So now we take the coefficient matrix, except where the 𝑥 column is, we replace it with the answer column.
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And now we repeat our steps to find the determinant of a three-by-three matrix.
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So we take one times the determinant of this matrix minus two times the determinant of these numbers plus negative two times the determinant of these numbers.
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So now we start evaluating just like we did before.
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And after simplifying, we get an answer of negative nine.
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So when solving for 𝑥, negative nine divided by negative three means 𝑥 is equal to three.
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So now let’s do the exact same process but with 𝑦.
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So we take the coefficient matrix but now instead of the 𝑦 column, replace it with the answer column.
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So we have three times the determinant of these numbers minus one times the determinant of these numbers plus negative two times the determinant of these numbers.
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So now we evaluate the determinants and then we multiply and simplify, and we get 75.
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And 75 divided by negative three means 𝑦 is negative 25.
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So lastly, 𝑧.
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Let’s go ahead and replace the 𝑧 column with the answer column.
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And now we evaluate.
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Three times the determinant of these numbers minus two times the determinant of these numbers plus one times the determinant of these numbers.
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So after evaluating, now we need to multiply and simplify, and we get 63.
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And 63 divided by negative three is negative 21.
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So after solving this system using determinants, 𝑥 equals three, 𝑦 equals negative 25, and 𝑧 equals negative 21.