Graph each equation of the system. Then solve the system to find the points of intersection. {

Graph each equation of the system. Then solve the system to...

Key Concepts

Graphing Functions

Graphing functions is the process of visually representing equations on a coordinate system. This is essential for understanding the behavior of different equations, identifying key features such as intercepts, slopes, and turning points, and analyzing how different functions relate to one another.

Square Root Functions

A square root function, such as y = ?x, has a distinctive curve that only exists for x ? 0. These functions are characterized by a gradual increase and are defined only for non-negative x-values, which highlights the importance of domain considerations when analyzing and solving equations involving square roots.

Linear Functions

Linear functions, like y = 6 - x, graph as straight lines with a constant rate of change. They are fundamental in mathematics due to their simplicity and the ease of predicting their behavior, including finding intercepts and understanding the relationship between variables through slope and y-intercept.

Systems of Equations

A system of equations consists of two or more equations that are solved together to find a set of variable values that satisfy all equations simultaneously. Solving systems graphically involves finding the point(s) where the graphs of the equations intersect, representing the common solution(s) to the system.

Domain Restrictions

Domain restrictions refer to the limitations on the set of input values for which a function is defined. In this context, the square root function is only valid for non-negative x-values, which must be considered when solving the system of equations to avoid extraneous solutions that do not meet the function’s inherent constraints.

Transcript

00:01 In this problem, we're asked to graph this system equation and solve it by finding their points of intersection.

00:08 So the first graph is a square root, and we know that we cannot have a square root negative number because that gives you non -real numbers.

00:17 So therefore, the x has to be started zero, and we know that square root zero is zero.

00:23 And then square one is one.

00:25 Squirrel four is two, and square root nine is three.

00:29 And this is the graph for the square root of the x.

00:34 And the next one we're going to graph is y equals 6 minus x, where 6 is your y intercept.

00:41 So you have to graph your y intercept.

00:43 And your slope is negative 1, so that's negative 1 over 1.

00:47 So that means you're going to go down 1 and then write 1, down 1, right 1.

00:53 So you can see that the points of intersection is located at 4 .2.

01:15 Okay, so now let's solve this and verify and verify our solution by solving this.

01:24 So because both equation equals to y, so i'm going to use substitution, then we know that the square root x is going to equal to 6 minus x.

01:33 And to get rid the square, i mean, to get rid the square root, you're going to square both sides.

01:39 So this is going to give us x equals to 6 minus x times 6 minus x.

01:47 That's what square means.

01:49 The 6 minus x is multiplied by itself.

01:53 So now you're going to foil and combine like terms...

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