Write a system of linear equations in three variables, and then use matrices to solve the system.
-The table below shows the number of birds for three selected years after an endangered species protection program was started. $$\begin{array} { l | c c c }
\mathrm { x } \text { (Number of years after 1980) } & 1 & 5 & 10 \
\hline \mathrm { y } \text { (Number of birds) } & 43 & 139 & 349
\end{array}$$ Use the quadratic function $y = a x ^ { 2 } + b x + c$ to model the data. Solve the system of linear equations involving $a , b$, and $c$ using matrices. Find the equation that models the data.
A) $y = 2 x ^ { 2 } + 12 x + 29$
B) $y = 3 x ^ { 2 } + 24 x + 24$
C) $y = 4 x ^ { 2 } - 12 x + 32$
D) $y = 4 x ^ { 2 } - 36 x + 25$

Question

Write a system of linear equations in three variables, and then use matrices to solve the system.
-The table below shows the number of birds for three selected years after an endangered species protection program was started. $$\begin{array} { l | c c c } 
\mathrm { x } \text { (Number of years after 1980) } & 1 & 5 & 10 \
\hline \mathrm { y } \text { (Number of birds) } & 43 & 139 & 349
\end{array}$$ Use the quadratic function $y = a x ^ { 2 } + b x + c$ to model the data. Solve the system of linear equations involving $a , b$, and $c$ using matrices. Find the equation that models the data.
A) $y = 2 x ^ { 2 } + 12 x + 29$
B) $y = 3 x ^ { 2 } + 24 x + 24$
C) $y = 4 x ^ { 2 } - 12 x + 32$
D) $y = 4 x ^ { 2 } - 36 x + 25$

Quizplus · Accepted Answer

To solve for $a$, $b$, and $c$, we substitute the given values of $x$ and $y$ into the quadratic equation $y = ax^2 + bx + c$. This gives us three equations:1. $43 = a(1)^2 + b(1) + c$2. $139 = a(5)^2 + b(5) + c$3. $349 = a(10)^2 + b(10) + c$Simplifying, we get:1. $43 = a + b + c$2. $139 = 25a + 5b + c$3. $349 = 100a + 10b + c$Using matrices to solve this system, we set up the augmented matrix and apply row reduction techniques to find $a$, $b$, and $c$. Solving the system yields $a = 2$, $b = 12$, and $c = 29$. Therefore, the equation that models the data is $y = 2x^2 + 12x + 29$.

Write a System of Linear Equations in Three Variables, and Then