Question

Graph each equation of the system. Then solve the system to find the points of intersection. $$ \left\\{\begin{array}{l} y=\sqrt{36-x^{2}} \\ y=8-x \end{array}\right. $$

Short answer

The points of intersection are \((4 + \sqrt{2}, 4 - \sqrt{2})\) and \((4 - \sqrt{2}, 4 + \sqrt{2})\).

Step-by-step solution

Identify the equations

The system of equations given is: 1. \(y = \sqrt{36 - x^2}\) 2. \(y = 8 - x\)

Graph the first equation

Graph the equation \(y = \sqrt{36 - x^2}\). This is the upper semicircle of a circle with radius 6 centered at the origin. The circle has the equation \(x^2 + y^2 = 36\).

Graph the second equation

Graph the equation \(y = 8 - x\). This is a straight line with a slope of -1 and y-intercept at 8 on the y-axis.

Find points of intersection

Find the points where the equations intersect by setting \(\sqrt{36 - x^2} = 8 - x\). Square both sides to remove the square root: \[ (\sqrt{36 - x^2})^2 = (8 - x)^2 \] \[ 36 - x^2 = 64 - 16x + x^2 \] Combine like terms to get a quadratic equation: \[ 2x^2 - 16x + 28 = 0 \]

Solve the quadratic equation

Solve the quadratic equation for x: \[ x^2 - 8x + 14 = 0 \] Using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -8\), and \(c = 14\): \[ x = \frac{8 \pm \sqrt{64 - 56}}{2} \] \[ x = \frac{8 \pm 2\sqrt{2}}{2} \] \[ x = 4 \pm \sqrt{2} \]

Find corresponding y values

Substitute \(x = 4 + \sqrt{2}\) and \(x = 4 - \sqrt{2}\) back into one of the original equations (e.g., \(y = 8 - x\)): For \(x = 4 + \sqrt{2}\): \[ y = 8 - (4 + \sqrt{2}) = 4 - \sqrt{2} \] For \(x = 4 - \sqrt{2}\): \[ y = 8 - (4 - \sqrt{2}) = 4 + \sqrt{2} \]

Write the points of intersection

The points of intersection are: \((4 + \sqrt{2}, 4 - \sqrt{2})\) and \((4 - \sqrt{2}, 4 + \sqrt{2})\).

Key concepts

graphing equations

Graphing equations is a fundamental skill used to visually represent equations on a coordinate plane. By plotting points that satisfy each equation and connecting them smoothly, you can understand the behavior of functions and see where they intersect. For example:

• A circle can be represented by the equation \(x^2 + y^2 = r^2\).
• A line can be represented by \(y = mx + c\), where m is the slope and c is the y-intercept.

When creating graphs, ensure to label the axes and scale them properly. Understanding the shape and positioning of these graphs can help in solving systems of equations by identifying their points of intersection.

quadratic formula

The quadratic formula is a powerful tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). The quadratic formula is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here:

• a is the coefficient of \(x^2\)
• b is the coefficient of \(x\)
• c is the constant term

Using this formula, you can find the roots of the quadratic equation by substituting the values of a, b, and c. Remember to take both the positive and negative values of the square root to find both potential solutions. The discriminant \(b^2 - 4ac\) helps determine the nature of the roots. If it's positive, there are two distinct real roots. If it's zero, there is exactly one real root. If it's negative, the roots are complex and conjugate pairs.

points of intersection

Points of intersection are crucial in solving systems of equations. These are the points where the graphs of different equations meet. To find these points:

• Set the equations equal to each other (for y-values).
• Solve for x.
• Substitute the x-values back into either original equation to find the y-values.

For example, given the equations \(y = \sqrt{36 - x^2}\) and \(y = 8 - x\), equate \(\sqrt{36 - x^2} = 8 - x\). Solve for x by squaring both sides to eliminate the square root. Then, simplify and solve the resulting quadratic equation. Once you have the x-values, substitute them back to find the corresponding y-values.

semi-circle graph

A semi-circle graph represents half of a circle, typically plotted from the equation of a circle. For example, the equation \(y = \sqrt{36 - x^2}\) represents the upper half of a circle with radius 6, centered at the origin \((0, 0)\). To understand this better:

• The complete circle equation can be written as \(x^2 + y^2 = 36\).
• By solving for y, you get \(y = \sqrt{36 - x^2}\) for the upper semi-circle and \(y = -\sqrt{36 - x^2}\) for the lower semi-circle.

This creates a semi-circle that spans from x = -6 to x = 6 on the x-axis, and from y = 0 to y = 6 on the y-axis. Plotting this helps visualize the context of the problem and finding intersection points when combined with other functions such as lines.

linear equations

Linear equations describe straight lines on a coordinate plane and are generally written as \(y = mx + c\), where:

• m is the slope of the line, indicating its steepness and direction.
• c is the y-intercept, indicating where the line crosses the y-axis.

For instance, the equation \(y = 8 - x\) represents a line with a y-intercept at 8 and a slope of -1. This means as the value of x increases by 1, the value of y decreases by 1. When graphing such a line, start by plotting the y-intercept and then use the slope to find additional points. Draw a straight line through these points to complete the graph. Linear equations are important in finding points of intersection with other types of graphs, such as semi-circles or parabolas.