Question - Solving a System of Equations using Elimination Method
Solution:
The image shows three different systems of simultaneous equations. Let's solve the first one (a) as requested:\[ \begin{align*}x - 3y - z &= 2 \\2x + 5y + 6z &= 3 \\3x + 2y + z &= 1 \\\end{align*} \]To solve this system, we can use the method of substitution or elimination. I will use the elimination method.First, let's multiply the third equation by 6 and subtract the second equation from it to eliminate $$ z $$.\[ \begin{align*}6(3x + 2y + z) &= 6(1) \\18x + 12y + 6z &= 6 \\\end{align*} \]Now subtract the second equation:\[ \begin{align*}18x + 12y + 6z - (2x + 5y + 6z) &= 6 - 3 \\16x + 7y &= 3 \quad \text{[Equation (4)]} \\\end{align*} \]Next, let's eliminate $$ z $$ by multiplying the first equation by 6 and the third equation by 1 and subtracting them:\[ \begin{align*}6(x - 3y - z) - (3x + 2y + z) &= 6(2) - 1 \\6x - 18y - 6z - 3x - 2y - z &= 12 - 1 \\3x - 20y - 7z &= 11 \quad \text{[Equation (5)]} \\\end{align*} \]Now we have two equations [Equation (4) and Equation (5)] with two unknowns, $$ x $$ and $$ y $$. Let's multiply Equation (4) by 20 and Equation (5) by 7 to eliminate $$ y $$:\[ \begin{align*}20(16x + 7y) &= 20(3) \\7(3x - 20y) &= 7(11) \\\end{align*} \]\[ \begin{align*}320x + 140y &= 60 \\21x - 140y &= 77 \\\end{align*} \]Adding these two equations together to eliminate $$ y $$:\[ \begin{align*}320x + 21x &= 60 + 77 \\341x &= 137 \\x &= \frac{137}{341} \\x &= \frac{1}{2.5} \\x &= 0.4 \\\end{align*} \]Now, plug the value of $$ x $$ back into Equation (4):\[ \begin{align*}16(0.4) + 7y &= 3 \\6.4 + 7y &= 3 \\7y &= 3 - 6.4 \\7y &= -3.4 \\y &= -\frac{3.4}{7} \\y &= -\frac{34}{70} \\y &= -\frac{17}{35} \\y &= -0.4857... \\\end{align*} \]Now that we have $$ x $$ and $$ y $$, we can solve for $$ z $$ using the first equation:\[ \begin{align*}0.4 - 3(-0.4857...) - z &= 2 \\0.4 + 1.4571 - z &= 2 \\z &= 0.4 + 1.4571 - 2 \\z &= -0.1429... \\\end{align*} \]Therefore, the solution to the system is $$ x \approx 0.4 $$, $$ y \approx -0.4857 $$, and $$ z \approx -0.1429 $$. Please note that the approximations for $$ y $$ and $$ z $$ are rounded.
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