Solve this system of equations:
[tex]\begin{array}{l}
y=x^2-3 x+12 \\
y=-2 x+14
\end{array}[/tex]
What are the solutions of the system of equations?
One solution is (-1, ).
The second solution (2, ).
Substitute the values of [tex]x[/tex], -1 and 2, into either original equation to solve for the values of [tex]y[/tex].
Answer (2)
Substitute x = − 1 into the equation y = − 2 x + 14 to find the corresponding y-value: y = − 2 ( − 1 ) + 14 = 16 .
Substitute x = 2 into the equation y = − 2 x + 14 to find the corresponding y-value: y = − 2 ( 2 ) + 14 = 10 .
The solutions to the system of equations are ( − 1 , 16 ) and ( 2 , 10 ) .
The solutions of the system of equations are ( − 1 , 16 ) and ( 2 , 10 ) , so the final answer is ( − 1 , 16 ) and ( 2 , 10 ) .
Explanation
Understanding the Problem We are given a system of two equations:
y = x 2 − 3 x + 12
y = − 2 x + 14
We are asked to find the solutions to the system, given that the x-coordinates of the solutions are -1 and 2. This means we need to find the corresponding y-coordinates for these x-values.
Finding the First Solution First, let's find the y-coordinate when x = − 1 . We can substitute this value into either equation. Let's use the second equation:
y = − 2 ( − 1 ) + 14 = 2 + 14 = 16
So, when x = − 1 , y = 16 . We can verify this by substituting x = − 1 into the first equation:
y = ( − 1 ) 2 − 3 ( − 1 ) + 12 = 1 + 3 + 12 = 16
Thus, the first solution is ( − 1 , 16 ) .
Finding the Second Solution Next, let's find the y-coordinate when x = 2 . Again, we can substitute this value into either equation. Let's use the second equation:
y = − 2 ( 2 ) + 14 = − 4 + 14 = 10
So, when x = 2 , y = 10 . We can verify this by substituting x = 2 into the first equation:
y = ( 2 ) 2 − 3 ( 2 ) + 12 = 4 − 6 + 12 = 10
Thus, the second solution is ( 2 , 10 ) .
Final Answer Therefore, the solutions to the system of equations are ( − 1 , 16 ) and ( 2 , 10 ) .
Examples
Systems of equations are used in many real-world applications. For example, they can be used to model supply and demand in economics, where the intersection of the supply and demand curves represents the equilibrium price and quantity. They are also used in physics to solve problems involving multiple forces or constraints, and in computer graphics to determine the intersection of lines and surfaces. Understanding how to solve systems of equations is a fundamental skill in many fields.
The solutions to the system of equations are (-1, 16) and (2, 10). Each solution was found by substituting the corresponding x-values into one of the original equations. Both solutions were verified by substituting back into the initial equations.
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