In Exercises 1–18, solve each system by the substitution method. { x+y=1 x^2+x y-y^2=-5

step 1 Hello, hope you're doing well. So we've got our two equations here that form our system of equat

In Exercises 1–18, solve each system by the substitution...

Key Concepts

Quadratic Equations

A quadratic equation is a type of polynomial equation that includes a variable squared and has the general form ax^2 + bx + c = 0. When applying the substitution method to certain systems of equations, the resulting equation can be quadratic. Understanding how to solve quadratic equations—through methods like factoring, completing the square, or the quadratic formula—is essential in finding all potential solutions.

Systems of Equations

A system of equations consists of multiple equations that share the same set of variables. The objective is to determine the values of these variables that satisfy all of the equations simultaneously. This concept forms the basis for analyzing relationships in algebra and many applied mathematics problems.

Substitution Method

The substitution method is an approach used to solve systems of equations by solving one equation for one variable and then substituting that expression into the other equation(s). This technique reduces the number of variables and transforms the system into a single equation, which is simpler to solve.

Transcript

00:01 Hello, hope you're doing well.

00:02 So we've got our two equations here that form our system of equations.

00:06 So in order to solve this system using substitution, you're going to take this first equation and solve for y, and then plug that in to our two y values here in the second equation.

00:17 So i'm going to rewrite our first equation here.

00:19 We're going to subtract x from both sides so that we can solve for y.

00:24 We're left with y is equal to 1 minus x.

00:27 All right, so now i'm going to rewrite our second equation over here.

00:30 We're going to plug in this equation, y equals 1 minus x for every occurrence of y.

00:37 So we got x squared plus x times instead of y.

00:42 We've got 1 minus x.

00:44 We got minus 1 minus x squared is equal to minus 5.

00:51 So i want to simplify this a little bit.

00:54 We've got x squared.

00:56 Distributing this x, we got x times 1, which is just going to be x.

01:00 With x times minus x, which is going to be minus x squared.

01:04 Now minus, we're going to square this one minus x.

01:08 We've got one squared.

01:09 It's just going to be one.

01:11 We've got one times minus x, just going to be minus x times another, or then minus another x.

01:18 That's going to be minus two x.

01:20 And this minus x squared, all squared here, just going to be plus x squared.

01:27 This is equal to minus five.

01:29 So we can't, since we've got a plus x squared and a minus x squared, we can just cancel these out.

01:33 We're left with x, and we're going to distribute this minus.

01:38 We've got minus 1, it's going to be minus 1.

01:41 Minus minus 2x is going to be plus 2x, and minus x squared, that's equal to minus x squared.

01:48 That's equal to minus 5.

01:50 So now we're going to move everything to the right -hand side of the equation.

01:55 So we're going to add x squared to those sides.

01:59 I'm going to add one to both sides.

02:03 These cancel combining this x and 2x.

02:06 We get 3x is equal to x squared minus 5 plus 1 is going to be x squared minus 4.

02:12 And subtracting 3x from both sides, we're left with 0 is equal to x squared minus 3x minus 4.

02:24 So now we want to factor this expression in order to make it easier to salt.

02:28 So we've got minus 4 times 1.

02:31 It's going to be minus 4...

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