Question

plot the graphs of both equations on the same coordinate plane. Find and label the points of intersection of the two graphs. $$ \begin{array}{l} y=x-1 \\ 2 x^{2}+3 y^{2}=12 \end{array} $$

Short answer

The intersection points of the line and ellipse are found by solving the quadratic equation \( 5x^2 - 6x - 9 = 0 \).

Step-by-step solution

Understand Each Equation

The first equation is a linear equation, which will graph as a straight line: \( y = x - 1 \). The second equation is a quadratic equation in two variables: \( 2x^2 + 3y^2 = 12 \), representing an ellipse.

Generate Points for the Line

For the line \( y = x - 1 \), we can generate points by choosing values for \( x \). For example, \((x=0, y=-1)\), \((x=1, y=0)\), and \((x=-1, y=-2)\). This will give you a straight line on the graph.

Rearrange and Simplify the Ellipse Equation

Rearrange the ellipse equation to make it easier for plotting. First divide the entire equation by 12 to get: \( \frac{x^2}{6} + \frac{y^2}{4} = 1 \). This is the standard form of an ellipse with semi-major and semi-minor axes.

Generate Points for the Ellipse

Use the rearranged ellipse equation to find points. For example, let \( x= \pm \sqrt{6} \), then \( y=0 \). Similarly, set \( y= \pm 2 \), then solve for \( x \).

Plot Both Graphs

On the same coordinate plane, plot the line using the points from Step 2 and plot the ellipse using points from Step 4. This will make it possible to visually see where the two graphs intersect.

Find Intersections Algebraically

Substitute the expression for \( y \) from the line equation into the ellipse equation: \( 2x^2 + 3(x-1)^2 = 12 \). Simplify and solve this equation for \( x \) to find the points of intersection.

Simplify the Substituted Equation

Expand \((x-1)^2\) to get \(x^2 - 2x + 1\), and substitute it back: \( 2x^2 + 3(x^2 - 2x + 1) = 12 \). This simplifies to \( 5x^2 - 6x + 3 = 12 \). Further simplify by moving 12 across: \( 5x^2 - 6x - 9 = 0 \).

Solve the Quadratic Equation

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve \( 5x^2 - 6x - 9 = 0 \). Here, \( a=5, b=-6, c=-9 \). This will give you the \( x \)-coordinates of the intersection points.

Calculate y-values of Intersections

For each \( x \) value from Step 8, substitute back into the line equation \( y = x - 1 \) to find the corresponding \( y \)-coordinate.

Verify and Label Intersection Points

After computing the intersection coordinates, verify that these points satisfy both original equations. If they do, label them on the graph as intersection points.

Key concepts

Ellipse and Line Intersection

In the world of mathematics, finding intersections between different curves is common. The intersection of an ellipse and a line is a frequent task, often posing a fascinating challenge. Let's break down what it means for these two different types of equations to intersect. An ellipse is a shape that looks like a flattened circle. It is represented by a quadratic equation in the form \( ax^2 + by^2 = c \), capturing its symmetrical nature. Meanwhile, a line is depicted through a linear equation, typically in the format \( y = mx + b \), showing a straight, direct path on a coordinate plane.

When these two equations are plotted together, they can intersect at one or more points. These intersections are places where the values of \( x \) and \( y \) satisfy both equations. To find these points, you generally solve the equations together, often involving solving a quadratic equation, which gives you the \( x \)-coordinates of the intersections. By substituting these \( x \) values back into the line equation, you can calculate the corresponding \( y \) values. Thus, you get the coordinates of intersection points that lie on both the ellipse and the line.

Coordinate Plane Plotting

Plotting graphs on a coordinate plane is like positioning points in a city map. The plane consists of two axes: horizontal (x-axis) and vertical (y-axis). Each point on this plane is determined by a pair of numbers called coordinates. For example, the point (2, 3) means from the origin (0, 0), you move 2 units along the x-axis and 3 units up along the y-axis.

When plotting the line \( y = x - 1 \), you start with choosing some x-values to find corresponding y-values from the equation. Points like (0, -1), (1, 0), and (-1, -2) give you a clear path to draw the straight line on the plane. For the ellipse, using the equation \( \frac{x^2}{6} + \frac{y^2}{4} = 1 \), you choose different pairs of x and y by considering the radii given by the coefficients. This way, you draw the shape of the ellipse around the coordinate plane, showing its symmetry and expansion. Both graphs are then plotted on the same plane to find any potential points of intersection.

Quadratic and Linear Equations

Quadratic and linear equations are two fundamental forms that depict different stories on a graph. A linear equation, as represented by \( y = mx + b \), describes a straight line. This line depicts a constant rate of change and is simple, like a highway connecting two places.

In contrast, a quadratic equation involves variables raised to the power of two and can be expressed as \( ax^2 + bx + c = 0 \) when simplified. When quadratic equations in two variables, such as an ellipse, are graphed, they form curves that are more dynamic, with no constant slope.

Interacting with these equations often means solving them together to find where their paths cross. This involves substituting the linear equation into the quadratic equation and simplifying it to find the solutions that satisfy both equations. These solutions often manifest as coordinates where the linear path meets the curvey, quadratic path. Understanding how to work with both types of equations is essential for mapping the interplay of lines and curves on the same graph.