What is the solution set to the following system?
x + y = 4
x^2 + y^2 = 16
Answers:
A. {(-4, 0)(0, 4)}
B. {(4, 0)(0, 4)}
C. {(4, 0)(0, -4)}
D. {(-4, 0)(0, -4)}
y = 4 - x
x^2 + (16 - 8 x + x^2) = 16
2 x^2 - 8 x = 0
x( 2x-8 ) = 0
x = 0 or x = 4
y = 0 or y = 4 of course, symmetry :)
(0,4) or (4,0)
Well, to find the solution set, let's first analyze this "circus" of an equation system.
The first equation is a simple straight line, while the second equation is the equation of a "ball" in the coordinate plane.
Now, we all know that clowns are excellent at juggling and balancing things. So let's find the points where this straight line balances perfectly with the ball.
Solving the first equation for y, we get y = 4 - x. Plugging this into the second equation gives us x^2 + (4 - x)^2 = 16.
Expanding and simplifying, we get x^2 + 16 - 8x + x^2 = 16. Combining like terms, we get 2x^2 - 8x = 0.
Factoring out 2x, we have 2x(x - 4) = 0.
So either 2x = 0 or (x - 4) = 0.
Hence, we have two possible solutions for x: x = 0 or x = 4.
Substituting these values of x back into the first equation, we find the corresponding y-values: when x = 0, y = 4; and when x = 4, y = 0.
Voila! We've found the solution set!
And the answer is... *drum roll*... B. {(4, 0)(0, 4)}.
So, remember, when life throws you a curveball and a straight line, always be ready to clown around and find the perfect balance!
To find the solution set to the given system of equations, we can solve it using the method of substitution. Here are the steps:
Step 1: Start with the first equation x + y = 4.
Step 2: Solve for x in terms of y by subtracting y from both sides of the equation:
x = 4 - y.
Step 3: Substitute this expression for x in the second equation x^2 + y^2 = 16:
(4 - y)^2 + y^2 = 16.
Step 4: Expand and simplify this equation:
16 - 8y + y^2 + y^2 = 16.
2y^2 - 8y = 0.
Step 5: Factor out 2y to solve for y:
2y(y - 4) = 0.
Step 6: Set each factor equal to zero and solve for y:
2y = 0 or y - 4 = 0.
y = 0 or y = 4.
Step 7: Substitute the values of y back into the expression for x to find the corresponding values:
For y = 0, x = 4 - 0 = 4.
For y = 4, x = 4 - 4 = 0.
Step 8: Write down the solution set as ordered pairs:
The solution set is {(4, 0), (0, 4)}.
Therefore, the correct answer is B. {(4, 0), (0, 4)}.
To find the solution set to the given system, we need to solve the system of equations simultaneously. To do this, let's go step by step.
1. Solve the first equation for one variable in terms of the other variable. In this case, it would be convenient to solve for x in terms of y:
x + y = 4
x = 4 - y
2. Substitute the expression for x in terms of y into the second equation:
(4 - y)^2 + y^2 = 16
3. Expand and simplify the equation:
(16 - 8y + y^2) + y^2 = 16
16 - 8y + y^2 + y^2 = 16
2y^2 - 8y = 0
4. Factor out common terms:
2y(y - 4) = 0
5. Set each factor equal to zero and solve for y:
y = 0 or y - 4 = 0
For y = 0, substitute this value back into x = 4 - y:
x = 4 - 0
x = 4
For y - 4 = 0, solve for y:
y = 4
Therefore, we have two sets of solutions: (x, y) = (4, 0) and (x, y) = (4, 0)
Now we can check which option matches our solution set:
A. {(-4, 0)(0, 4)} - This option does not match our solution set as it includes (-4, 0) and (0, 4) instead of (4, 0).
B. {(4, 0)(0, 4)} - This option matches our solution set: (x, y) = (4, 0) and (x, y) = (0, 4).
C. {(4, 0)(0, -4)} - This option does not match our solution set as it includes (0, -4) instead of (0, 4).
D. {(-4, 0)(0, -4)} - This option does not match our solution set as it includes (-4, 0) and (0, -4) instead of (4, 0).
Therefore, the correct answer is B. {(4, 0)(0, 4)}.