Question Statement

Asad is arranging a flash light in a marriage ceremony. The position of the flash light is at the intersection of the lines:

• $2x+y-23=0$
• $0.5x-y+3=0$
• $x-y=1$

Find the position of the flashlight (i.e., the point of intersection of these three lines).

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Background and Explanation

Three lines are said to be concurrent if they all pass through a single common point. To find this intersection point, we can verify that the lines meet at one point using the determinant condition for concurrency, then solve any two of the equations simultaneously to find the exact coordinates.

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Solution

The three given lines are: \begin{align} 2x + y - 23 &= 0 \\ 0.5x - y + 3 &= 0 \\ x - y - 1 &= 0 \end{align}

Step 1: Verify that the lines are concurrent

To check if all three lines meet at a single point, we calculate the determinant of the coefficient matrix:

$\Delta =\left|\begin{array}{ccc}2& 1& -23\\ 0.5& -1& 3\\ 1& -1& -1\end{array}\right|$

Expanding along the first row:

$\begin{array}{rl}\Delta & =2\left|\begin{array}{cc}-1& 3\\ -1& -1\end{array}\right|-1\left|\begin{array}{cc}0.5& 3\\ 1& -1\end{array}\right|+(-23)\left|\begin{array}{cc}0.5& -1\\ 1& -1\end{array}\right|\\ & =2[(-1)(-1)-(3)(-1)]-1[(0.5)(-1)-(3)(1)]-23[(0.5)(-1)-(-1)(1)]\\ & =2(1+3)-1(-0.5-3)-23(-0.5+1)\\ & =2(4)-1(-3.5)-23(0.5)\\ & =8+3.5-11.5\\ & =0\end{array}$

Since $\Delta =0$, the three lines are concurrent (they intersect at exactly one common point).

Step 2: Find the point of intersection

To find the coordinates of the flashlight, we solve any two of the three equations. Let's solve equations (i) and (iii).

Adding equations (i) and (iii) to eliminate $y$:

$\begin{array}{rcl}(2x+y-23)+(x-y-1)& =& 0\\ 3x-24& =& 0\\ 3x& =& 24\\ x& =& 8\end{array}$

Substitute $x=8$ into equation (iii):

$\begin{array}{rcl}8-y-1& =& 0\\ 7-y& =& 0\\ y& =& 7\end{array}$

Thus, the position of the flashlight is at the point $(8,7)$.

Verification: We can check that $(8,7)$ satisfies equation (ii) as well: $0.5(8)-7+3=4-7+3=0✓$

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Key Formulas or Methods Used

• Condition for concurrency of three lines: For lines $a_{1}x+b_{1}y+c_1=0$, $a_{2}x+b_{2}y+c_2=0$, and $a_{3}x+b_{3}y+c_3=0$, the condition is: $\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|=0$

• Elimination method for simultaneous equations: Add or subtract equations to eliminate one variable and solve for the other

• Substitution method: Substitute the known value of one variable back into an equation to find the second variable

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Summary of Steps

1. Identify the three line equations and label them (i), (ii), and (iii)
2. Verify concurrency by calculating the $3\times 3$ determinant of coefficients; confirm it equals zero
3. Select any two equations (e.g., (i) and (iii)) and add them to eliminate the $y$-variable
4. Solve for $x$: $3x=24\Rightarrow x=8$
5. Substitute back to find $y=7$
6. State the final position as the coordinate point $(8,7)$