A system of equations is a set of two or more equations involving the same set of variables. Solving these systems means finding the values of the variables that satisfy all equations simultaneously.
General Form:
\[
\begin{aligned}
a_1x + b_1y &= c_1 \\
a_2x + b_2y &= c_2
\end{aligned}
\]
where: a and b are coefficients and c is a constant term independent of any variable
In engineering and applied sciences, systems of equations model real-world relationships such as force balance, circuit laws, material costs, and more.
Methods of Solving:
Substitution: Solve one equation for one variable and substitute it into the other.
Elimination (or Addition Method): Add or subtract equations to eliminate a variable, then solve the resulting equation.
Graphical Method: Graph each equation and identify the point(s) of intersection.
Matrix Method (for larger systems): Convert the system to matrix form and solve using row operations or inverse matrices.
Matrix Form:
For larger systems, the equations can be written in matrix form as:
\[
A\vec{x} = \vec{b}
\]
Where:
$A$ is the coefficient matrix
$\vec{x}$ is the column vector of variables
$\vec{b}$ is the constant column vector
Solution Types:
Unique Solution: The system has one set of values for the variables (intersect at one point).
Infinitely Many Solutions: The equations represent the same line or plane.
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Exam Generator Problems
Additional board-style practice items for this topic.
Given the following matrices: $A = [2, -3, 4; 1, -1, 7; 5, 6, 8]$ and $B = [3; 4; 9]$.
What is the determinant of the matrix A?
-64
64
-137
137
What is the cofactor of the 7 entry of matrix A?
27
-27
32
-32
What is the product A & B?
[36; 24; 84]
[26; 38; 8]
[15; 52; 96]
[30; 62; 111]
Part 1.
Expand the determinant of $A=\begin{bmatrix}2&-3&4\\1&-1&7\\5&6&8\end{bmatrix}$ along the first row: $|A|=2[(-1)(8)-7(6)]-(-3)[1(8)-7(5)]+4[1(6)-(-1)(5)]$ $=2(-50)+3(-27)+4(11)=-137$. $\boxed{-137}$
Part 2.
The entry 7 is in row 2, column 3. Its cofactor is: $C_{23}=(-1)^{2+3}\begin{vmatrix}2&-3\\5&6\end{vmatrix}$ $=-(2(6)-(-3)(5))=-(12+15)=-27$. $\boxed{-27}$
Part 3.
$A B=\begin{bmatrix}2&-3&4\\1&-1&7\\5&6&8\end{bmatrix}\begin{bmatrix}3\\4\\9\end{bmatrix}$ First row: $2(3)-3(4)+4(9)=30$ Second row: $1(3)-1(4)+7(9)=62$ Third row: $5(3)+6(4)+8(9)=111$ $\boxed{[30; 62; 111]}$
Given the Matrix A and its inverse matrix B: $A = [2, 1, 3; 0, -1, 2; 4, 3, 1]$, $B = 1/6 \times [x, 8, y; 8, -10, -4; 4, z, -2]$.
Determine the value of $x$.
-2
5
1
-7
Determine the value of $y$.
1
5
-7
-2
Determine the value of $z$.
-2
-7
5
1
Part 1.
Since $B$ is the inverse of $A$, $A\left(6B\right)=6I$. Let $6B=\begin{bmatrix}x&8&y\\8&-10&-4\\4&z&-2\end{bmatrix}$. Use the first row of $A$ and the first column of $6B$: $[2,1,3]\cdot[x,8,4]=6$ $2x+8+12=6 \Rightarrow 2x=-14 \Rightarrow x=-7$. $\boxed{-7}$
Part 2.
Use $A(6B)=6I$. The first row, third column entry must be 0: $[2,1,3]\cdot[y,-4,-2]=0$ $2y-4-6=0 \Rightarrow 2y=10 \Rightarrow y=5$. $\boxed{5}$
Part 3.
Use $A(6B)=6I$. The second row, second column entry must be 6: $[0,-1,2]\cdot[8,-10,z]=6$ $10+2z=6 \Rightarrow 2z=-4 \Rightarrow z=-2$. $\boxed{-2}$
Find the eigenvalues associated with the matrix $\begin{bmatrix} 3 & 6 \ 1 & 4 \end{bmatrix}$.
$3$ or $6$
$1$ or $4$
$3$ or $4$
$6$ or $1$
Eigenvalues satisfy $\det(A-\lambda I)=0$. $\begin{vmatrix}3-\lambda&6\\1&4-\lambda\end{vmatrix}=0$ $(3-\lambda)(4-\lambda)-6=0$ $\lambda^2-7\lambda+6=0$ $(\lambda-6)(\lambda-1)=0$, so $\lambda=6$ or $1$. $\boxed{6\text{ or }1}$
Question Bank: w8
MSTE - Algebra / Simultaneous Equations / MSTE May 2019
Solve for $x$ from the following equations: $xy = 12$; $yz = 20$; $zx = 15$.
±3
±5
±4
±2
Multiply $xy$ and $zx$, then divide by $yz$: $\frac{(xy)(zx)}{yz} = x^2 = \frac{12 \times 15}{20} = 9$ $\boxed{x = \pm 3}$
