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.
-->
🧭 Jump to:
Scroll to zoom
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 $yz$: $\frac{(xy)(zx)}{yz} = x^2 = \frac{12 \times 15}{20} = 9$ $\boxed{x = \pm 3}$
Question Bank: w248
MSTE - Algebra / Simultaneous Equations / MSTE April 2025
A rectangular waterfront lot has a perimeter of 1000 feet. To create a sense of privacy, the lot's owner decides to fence along three sides, excluding the side that fronts the water. An expensive fencing along the lot's front length costs Php25 per foot, and an inexpensive fencing along two side widths costs only Php5 per foot. The total cost of the fencing along all three sides comes to Php9500. What is the lot's dimensions?
300' × 100'
400' × 100'
400' × 200'
300' × 200'
Let $x$ be the front length and $y$ the side width.
Perimeter: $2x + 2y = 1000$ (1)
Cost of fencing (one front length at Php25/ft plus two widths at Php5/ft): $25x + 5(2y) = 9500$ (2)
MSTE - Algebra / Simultaneous Equations / MSTE April 2025
A rectangular holding pen for cattle is to be designed so that its perimeter is 92 feet and its area is 525 sq. feet. Find the dimensions of the holding pen.
