Question 10.4.5: Using Cramer’s Rule Use Cramer’s Rule to solve the system of...
Using Cramer’s Rule
Use Cramer’s Rule to solve the system of equations.
\left\{\begin{array}{r}7 x+y+z-1=0 \\5 x+3 z-2 y=4 \\4 x-z+3 y=0\end{array}\right.
STUDY TIP
When you apply Cramer’s Rule, make sure that all linear equations are written in the form
ax + by + cz = k.
First, rewrite the system of equations so that the terms on the left side of the equal signs are in the proper order and only the constant terms appear on the right side.
\left\{\begin{array}{l}7 x+y+z=1 \\5 x-2 y+3 z=4 \\4 x+3 y-z=0\end{array}\right.
Step 1
D=\left|\begin{array}{rrr}7 & 1 & 1 \\5 & -2 & 3 \\4 & 3 & -1\end{array}\right| D is the determinant of the coefficient matrix.
=7\left|\begin{array}{rr}-2 & 3 \\3 & -1\end{array}\right|-5\left|\begin{array}{rr}1 & 1 \\3 & -1\end{array}\right|+4\left|\begin{array}{rr}1 & 1 \\-2 & 3\end{array}\right| Expand by column 1.
=7(2-9)-5(-1-3)+4(3+2) Evaluate determinants of order 2.
=7(-7)-5(-4)+4(5)=-9 Simplify.
Because D ≠ 0, the system has a unique solution.
Step 2 D_{x}=\left|\begin{array}{rrr}1 & 1 & 1 \\4 & -2 & 3 \\0 & 3 & -1\end{array}\right| Replace the x-coefficients in the first column in D with the constants.
=1\left|\begin{array}{rr}-2 & 3 \\3 & -1\end{array}\right|-4\left|\begin{array}{rr}1 & 1 \\3 & -1\end{array}\right|+0\left|\begin{array}{rr}1 & 1 \\-2 & 3\end{array}\right| Expand by Column 1.
=1(2-9)-4(-1-3) Evaluate determinants of order 2.
=1(-7)-4(-4)=9 Simplify.
Step 3 D_{y}=\left|\begin{array}{rrr}7 & 1 & 1 \\5 & 4 & 3 \\4 & 0 & -1\end{array}\right| Replace column 2 in D with the constants.
=-1\left|\begin{array}{rr}5 & 3 \\4 & -1\end{array}\right|+4\left|\begin{array}{rr}7 & 1 \\4 & -1\end{array}\right|-0\left|\begin{array}{ll}7 & 1 \\5 & 3\end{array}\right| Expand by column 2 because it contains a zero.
=-1(-5-12)+4(-7-4) Evaluate the determinants.
= 17 – 44 = -27 Simplify.
Step 4 D_{z}=\left|\begin{array}{rrr}7 & 1 & 1 \\5 & -2 & 4 \\4 & 3 & 0\end{array}\right| Replace column 3 in D with the constants.
=1\left|\begin{array}{rr}5 & -2 \\4 & 3\end{array}\right|-4\left|\begin{array}{ll}7 & 1 \\4 & 3\end{array}\right|+0\left|\begin{array}{rr}7 & 1 \\5 & -2\end{array}\right| Expand by column 3.
=(15+8)-4(21-4) Evaluate the determinants.
=23-4(17)=-45 Simplify.
Step 5 Cramer’s Rule gives the following values:
x=\frac{D_{x}}{D}=\frac{9}{-9}=-1 From Steps 1 and 2
y=\frac{D_{y}}{D}=\frac{-27}{-9}=3 From Steps 1 and 3
z=\frac{D_{z}}{D}=\frac{-45}{-9}=5 From Steps 1 and 4
Therefore, the solution set is \{(-1,3,5)\}.
Check: You should check the solution.