In Problems 9-18, verify that the values of the variables listed are solutions of the system of

step 1 Here, we're going to try to solve the system linear equations. 9y minus 18x equals 5, and 2x min

In Problems 9-18, verify that the values of the variables...

Key Concepts

Systems of Linear Equations

A system of linear equations consists of two or more equations involving the same set of variables. The solutions to these systems are the values of the variables that satisfy every equation simultaneously. Understanding such systems is fundamental because they model many real-world situations and can be solved using various methods such as substitution, elimination, or matrix operations.

Solution Verification by Substitution

Solution verification by substitution involves replacing the variables in each equation of a system with the provided candidate values. If every equation is satisfied (i.e., both sides of the equation are equal after substitution), then the candidate values constitute a valid solution for the system. This concept is essential to ensure the correctness of a proposed solution in any mathematical context.

Transcript

00:01 Here, we're going to try to solve the system linear equations.

00:03 9y minus 18x equals 5, and 2x minus y equals 3 by a substitution.

00:08 So the first step is to get the x or y by itself in either equation.

00:12 So just try to get the x or y by itself.

00:14 It doesn't matter which equation you use.

00:15 I'm going to choose this one because it's got smaller numbers.

00:18 So let's use that plus the coefficient of y is a negative 1.

00:22 Whenever the coefficient of the variable you're isolating is a 1 or a negative 1, it makes it a little easier.

00:28 So let's solve that for y.

00:31 You get 2x minus y is equal to 3.

00:34 Your goal is to get the y by itself.

00:37 So let's add y to both sides.

00:41 And if we're going to bring the y is to the right, let's get rid of this 3 from the right and move it to the left.

00:45 So how do you undo plus 3 minus 3 to both sides? these are not like terms, so they can't be combined.

00:53 Therefore, we have our answer.

00:54 Y must be equal to.

00:56 This is gone, right? this is a zero pair.

00:59 And this over here is gone.

01:01 This is a zero pair, right? so you just get y is equal to 2x minus 3.

01:07 Once we know that, we've got our value of y that we can use to replace y in the other equation.

01:12 So now we're going to substitute the x or y value.

01:15 In this case we have the y value 2x minus 3 into the other equation.

01:19 So we're going to replace this y right here with 2x minus 3 and solve for x.

01:27 And this will give us the x location of the intersection.

01:30 So this is 9 times the quantity of y, which in our case is 2x minus 3.

01:38 And then we're going to take away 18x from that, and that's equal to 5.

01:44 So 9 times the quantity of 2x minus 18x equals 5 and solve it.

01:49 Distribute 9 times 2x is 18 x is 18 x is 9 times negative 3 is negative 27.

01:56 So really you've got 18x minus 27 minus 18x is equal to 5.

02:00 Now, what you'll notice is when you combine your like terms here, that x goes away.

02:05 So you get negative 27 is equal to 5.

02:08 And last time i checked, that's not true, right? that's false.

02:11 Which means that we think there is no solution.

02:16 Now, what does that mean? it means there's no x location where these two lines intersect.

02:21 There's no ordered pair that is in common between these two lines, right? so that means there's no order pair x and y that's going to work...

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