Questions: Solve the system graphically. (If there is no solution, enter NO SOLUTION.) x+y=10 x^2+y^2-10x=0 (x, y)=( )(smaller x-value) (x, y)=( )(larger x-value)

Solve the system graphically. (If there is no solution, enter NO SOLUTION.) x+y=10 x^2+y^2-10x=0 (x, y)=( )(smaller x-value) (x, y)=( )(larger x-value)
Transcript text: Solve the system graphically. (If there is no solution, enter NO SOLUTION.) \[ \begin{array}{c} \left\{\begin{array}{c} x+y=10 \\ x^{2}+y^{2}-10 x=0 \end{array}\right. \\ (x, y)=(\square)(\text { smaller } x \text {-value }) \\ (x, y)=(\square)(\text { larger } x \text {-value }) \end{array} \]
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Solution

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Solution Steps

To solve the system graphically, we need to plot the two equations and find their intersection points. The first equation is a linear equation, and the second equation is a circle. The intersection points of these two graphs will give us the solutions.

Step 1: Rewrite the Equations

First, we rewrite the given system of equations for clarity: \[ \begin{cases} x + y = 10 \\ x^2 + y^2 - 10x = 0 \end{cases} \]

Step 2: Solve the First Equation for \( y \)

From the first equation, solve for \( y \): \[ y = 10 - x \]

Step 3: Substitute \( y \) into the Second Equation

Substitute \( y = 10 - x \) into the second equation: \[ x^2 + (10 - x)^2 - 10x = 0 \]

Step 4: Simplify the Equation

Expand and simplify the equation: \[ x^2 + (10 - x)^2 - 10x = 0 \] \[ x^2 + (100 - 20x + x^2) - 10x = 0 \] \[ 2x^2 - 30x + 100 = 0 \]

Step 5: Solve the Quadratic Equation

Solve the quadratic equation \( 2x^2 - 30x + 100 = 0 \) using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): \[ a = 2, \quad b = -30, \quad c = 100 \] \[ x = \frac{30 \pm \sqrt{(-30)^2 - 4 \cdot 2 \cdot 100}}{2 \cdot 2} \] \[ x = \frac{30 \pm \sqrt{900 - 800}}{4} \] \[ x = \frac{30 \pm \sqrt{100}}{4} \] \[ x = \frac{30 \pm 10}{4} \] \[ x = \frac{40}{4} = 10 \quad \text{or} \quad x = \frac{20}{4} = 5 \]

Step 6: Find Corresponding \( y \) Values

For \( x = 10 \): \[ y = 10 - 10 = 0 \] For \( x = 5 \): \[ y = 10 - 5 = 5 \]

Final Answer

The solutions to the system are: \[ (x, y) = (5, 5) \quad \text{(smaller \( x \)-value)} \] \[ (x, y) = (10, 0) \quad \text{(larger \( x \)-value)} \]

\[ \boxed{(x, y) = (5, 5)} \] \[ \boxed{(x, y) = (10, 0)} \]

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