Find the solution set for each system by graphing both of the system's equations in the same rec

Find the solution set for each system by graphing both of...

Key Concepts

Systems of Equations

A system of equations consists of two or more equations that are solved together, meaning the solution is a set of values that satisfies every equation in the system at the same time. This concept is fundamental in algebra and is frequently employed to find points where two or more graphs intersect.

Graphing Equations in the Rectangular Coordinate System

Graphing equations involves plotting curves or lines on a Cartesian plane, which is defined by horizontal and vertical axes. This method visually represents the relationships between variables and is particularly useful for finding the intersection points of different equations.

Intersection of Graphs

The intersection of graphs refers to the points where two or more curves or lines cross each other on the coordinate system. These intersection points represent the common solutions to the corresponding equations and are key to solving systems graphically.

Quadratic Equations and Their Graphs

Quadratic equations, typically represented in the form ax² + bx + c = 0, graph as parabolas on the Cartesian plane. Understanding the properties of parabolas such as vertex, axis of symmetry, and direction of opening is essential for analyzing how they might intersect with other types of graphs, like straight lines.

Verification of Solutions

Verification of solutions involves substituting the found values back into the original equations to ensure they satisfy all given conditions. This step is crucial, especially for equations that involve transformations like squaring, to avoid extraneous solutions.

Transcript

00:01 Okay, for this problem, we are asked to find the solution set of these two equations by graphing them on the same coordinate plane and then finding their points of intersection.

00:10 Then once we find their points of intersection, we're going to check those solutions and just by plugging them back into those equations that we have up there, the green and in the blue.

00:20 So first up, we want to start off by graphing the equation that i have in the blue because it is the simplest.

00:28 So as we can see, we have an mx plus b.

00:35 We can add a little plus zero here.

00:40 So the m is telling us our slope and the b is telling us our y intercept.

00:45 So we can confidently say that it's zero, zero.

00:48 It is crossing that x -axis.

00:51 And it's going to go down 1.

00:56 So we're at negative 1 and over 2.

01:01 Then we're going to continue the same process, so down 1 over 2.

01:11 And now that we've made enough points, we're just going to go ahead and draw a line through those.

01:18 Now the trickier part is our green equation, so we're just going to quickly foil this out.

01:35 So we get this.

01:36 So now our 4s can cancel out, and we're left with y minus 4, or y squared minus 4 y equals x.

01:44 This looks really close, really similar as if it was x squared plus something, and it is a parabola, but our parabola is going to be kind of looking like this on our coordinate plane.

01:59 So we can factor this just like we would a parabola to find our zeros.

02:08 So y equals zero and y equals four.

02:13 So that is where we're going to plot some of our variables.

02:15 Points, so at 0 ,0, and at 04 is where our y intercepts are going to be.

02:24 And now we need to remember a little equation for how we can find the point of the parabola.

02:36 So we need negative b over 2a.

02:43 So in our, in this equation, we have our, our, in this equation, we have our, our, a, oops, we have our a and we have our b.

02:56 So once we plug those in, our negative b is, so our b is negative 4, so negative negative is going to be 4 over 2 times, that's 1 times y squared.

03:08 So it's going to be that.

03:09 So that's 4 over 2, which is going to be 2.

03:13 And so that is going to be the y coordinate of that point...

Please give Ace some feedback