1D - Simultaneous linear equations
2D - Content video: Solving simultaneous linear equations
This video covers the theory and several examples relating to how to solve simultaneous linear equations.
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Introduction to simultaneous linear equations:
A set of simultaneous linear equations involves two, or more, linear relationships. For example: \(y=2x+1\) and \(x+y=4\).
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How many solutions?
When solving two linear equations simultaneously it is possible to have no solutions, one solution or infinitely many solutions.
When solving two linear equations simultaneously it is possible to have no solutions, one solution or infinitely many solutions.
No solutions:
This occurs when two linear equations are parallel, but are not the same line.
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One solution:
This occurs when the two linear equations are non-parallel.
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Infinitely many solutions:
This occurs when the two lines are exactly the same.
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1D - Example 1: Solving simultaneous linear equations using substitution
Solve the following pair of simultaneous equations using substitution:
(1) \(4x-y=-12\) (2) \(2x+5y=16\) |
1D - Example 1: Video solution
Video solution coming soon! \[x=-2,y=4\] 1D - Example 1: Practice
Question 1: ABC Question 2: ABC 1D - Example 1: Solutions
Question 1: ABC Question 2: ABC |
1D - Example 2: Solving simultaneous linear equations using elimination
Solve the following pair of simultaneous equations using elimination:
(1) \(2x-3y=9\) (2) \(2x+y=13\) |
1D - Example 2: Video solution
Video solution coming soon! \[x=6,y=1\] 1D - Example 2: Practice
Question 1: ABC Question 2: ABC 1D - Example 2: Solutions
Question 1: ABC Question 2: ABC |
1D - Example 3: Solving simultaneous literal linear equations
Solve the following pair of simultaneous equations, for \(x\) and \(y\), using an appropriate method:
(1) \(ax+by=c\) (2) \(bx+ay=-c\) |
1D - Example 3: Video solution
Video solution coming soon! \[x=\frac{c}{a-b},y=\frac{-c}{a-b}\] 1D - Example 3: Practice
Question 1: ABC Question 2: ABC 1D - Example 3: Solutions
Question 1: ABC Question 2: ABC |
1D - Example 4: Solving simultaneous linear equations (CAS)
Using a CAS calculator, determine the solution to the simultaneous equations
(1) \(x+7y=10\) (2) \(3x-2y=7\) |
1D - Example 4: Video solution
Video solution coming soon! \[x=3,y=1\] 1D - Example 4: Practice
Question 1: ABC Question 2: ABC 1D - Example 4: Solutions
Question 1: ABC Question 2: ABC |
Using matrices with simultaneous linear equations:
As previously stated, for simultaneous linear equations it is possible to have no solution, one solution or infinitely many solutions. The system of simultaneous equations, shown below, can be represented in matrix form \(AX=B\): \[\left\{\begin{matrix}
ax+by=e\\
cx+dy=f
\end{matrix}\right.
\Rightarrow
\begin{bmatrix}
a & b\\
c & d
\end{bmatrix}
\begin{bmatrix}
x\\
y
\end{bmatrix}
=
\begin{bmatrix}
e\\
f
\end{bmatrix}\] The system described above will have
ax+by=e\\
cx+dy=f
\end{matrix}\right.
\Rightarrow
\begin{bmatrix}
a & b\\
c & d
\end{bmatrix}
\begin{bmatrix}
x\\
y
\end{bmatrix}
=
\begin{bmatrix}
e\\
f
\end{bmatrix}\] The system described above will have
- One unique solution if \(det(A)\neq 0\)
- No solutions OR infinitely many solutions if \(det(A)=0\)
1D - Example 5: The number of solutions for simultaneous linear equations (VCAA, 2007)
[VCAA, 2007 Exam 2 MC Question 5]
The simultaneous linear equation \(mx+12y=24\) and \(3x+my=m\) have a unique solution for A. \(m=6\) or \(m=-6\) B. \(m=12\) or \(m=3\) C. \(m\in R\) \ { \(-6,6\) } D. \(m=2\) or \(m=1\) E. \(m\in R\) \ { \(-12,-3\) } |
1D - Example 5: Video solution
Video solution coming soon! Option C is correct. 1D - Example 5: Practice
Question 1: For what value(s) of \(p\) will the linear equations \(y-4x=3\) and \(2y-px=-8\) have one point of intersection? 1D - Example 5: Solutions
Question 1: \[\left\{\begin{matrix} -4x+y=3\\ -px+2y=-8 \end{matrix}\right. \Rightarrow \begin{bmatrix} -4 & 1\\ -p & 2 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} 3\\ -8 \end{bmatrix}\] \[\therefore (-4)(2)-(1)(-p)\neq 0\]Therefore, one unique solution for \(p\neq 8\) |
1D - Example 6: The number of solutions for simultaneous linear equations (VCAA, 2006)
[VCAA, 2006 Exam 2 MC Question 19]
The simultaneous linear equations \((m-2)x+3y=6\) and \(2x+(m-3)y=m-1\) have no solutions for A. \(m\in R\) \ { \(0,5\) } B. \(m\in R\) \ { \(0\) } C. \(m\in R\) \ { \(6\) } D. \(m=5\) E. \(m=0\) |
1D - Example 6: Video solution
Video solution coming soon! Option E is correct. 1D - Example 6: Practice
Question 1: ABC Question 2: ABC 1D - Example 6: Solutions
Question 1: ABC Question 2: ABC |
1D - Example 7: The number of solutions for simultaneous linear equations (VCAA, 2008)
[VCAA, 2008 Exam 2 MC Question 6]
The simultaneous linear equations \(ax+3y=0\) and \(2x+(a+11)y=0\) where \(a\) is a real constant, have infinitely many solutions for A. \(a\in R\) B. \(a\in\) { \(-3,2\) } C. \(a\in R\) \ { \(-3,2\) } D. \(a\in\) { \(-2,3\) } E. \(a\in R\) \ { \(-2,3\) } |
1D - Example 7: Video solution
Video solution coming soon! Option B is correct. 1D - Example 7: Practice
Question 1: ABC Question 2: ABC 1D - Example 7: Solutions
Question 1: ABC Question 2: ABC |