Question

Graph each equation of the system. Then solve the system to find the points of intersection. $$ \left\\{\begin{aligned} y &=3 x-5 \\ x^{2}+y^{2} &=5 \end{aligned}\right. $$

Short answer

Points of intersection: (1, -2) and (2, 1).

Step-by-step solution

Graph the Linear Equation

Graph the first equation, which is a linear equation: \( y = 3x - 5 \). To graph it, find the y-intercept and slope. The y-intercept is -5 (where the line crosses the y-axis), and the slope is 3 (rise over run).

Graph the Circle

Graph the second equation, which represents a circle: \( x^2 + y^2 = 5 \). To graph it, identify the center and the radius. The equation corresponds to a circle centered at (0, 0) with a radius of \( \sqrt{5} \).

Solve for Intersection Points

Set the two equations equal to each other by substituting the linear equation into the circle equation. Replace y in the circle equation: \( x^2 + (3x - 5)^2 = 5 \). Expand and solve for x. \(x^2 + 9x^2 - 30x + 25 = 5 \), \(10x^2 - 30x + 20 = 0 \), \( x^2 - 3x + 2 = 0 \), Solve for x using the quadratic formula: \( x = 1 \text{ or } x = 2 \).

Find Corresponding y Values

For each x value found, substitute it back into the linear equation to find the corresponding y values. For x = 1, \( y = 3(1) - 5 = -2 \). For x = 2, \( y = 3(2) - 5 = 1 \). So, the points of intersection are (1, -2) and (2, 1).

Key concepts

Graphing linear equations

Graphing linear equations is essential when solving systems of equations. Linear equations are in the form of \( y = mx + b \), where \(m\) is the slope and \(b\) is the y-intercept. To graph a linear equation:

• Start with the y-intercept. This is where the line crosses the y-axis.
• Next, use the slope. The slope is often given as a fraction, \( \frac{rise}{run} \). From the y-intercept, move up or down (rise) and then left or right (run) according to the slope.
• Draw the line through these points.

For example, with \( y = 3x - 5 \), the y-intercept is -5. Start at (0, -5) and plot additional points based on the slope. With a slope of 3, move up 3 units and right 1 unit. Continue plotting points and draw the line through them.

Graphing circles

Circles are graphed using the general form of the circle equation \( (x - h)^{2} + (y - k)^{2} = r^{2} \), where \( (h, k) \) is the center of the circle and \( r \) is the radius. To graph a circle:

• Identify the center \( (h, k) \). In the example \( x^{2} + y^{2} = 5 \), the center is (0,0).
• Determine the radius \( r \) by taking the square root of the constant on the right side of the equation. Here, \( r = \sqrt{5} \).
• Draw the circle with the center at (0,0) and \( \sqrt{5} \) as the radius.

Graphing circles helps in visualizing where they intersect with other graphs like lines.

Intersection points

Finding intersection points is crucial for solving systems of equations. Intersection points are where the graphs of the equations meet. These are common solutions to both equations. To find intersection points:

• Graph each equation on the same set of axes.
• Look for points where the graphs cross.
• Solve algebraically by setting equations equal to each other and solving for the variables.

In this case, set \( y = 3x - 5 \) into \( x^{2} + y^{2} = 5 \). Substitute to find \( x \) values, then use these \( x \) values to find corresponding \( y \) values. This will give you the intersection points.

Quadratic formula

The quadratic formula is a powerful tool for solving quadratic equations of the form \( ax^{2} + bx + c = 0 \). The formula is: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \]. To use the quadratic formula:

• Identify coefficients \( a, b, \text{ and } c \) from the equation.
• Substitute them into the quadratic formula.
• Simplify under the square root \( b^{2} - 4ac \). This is called the discriminant.
• Calculate the solutions using the \pm sign to get two possible values for \( x \).

For example, with \( x^{2} - 3x + 2 = 0 \), apply the quadratic formula to find \( x = 1 \text{ or } x = 2 \). Use these values to find corresponding \( y \) values in the original linear equation, providing the intersection points.