Solve each system by substitution. 7 x+6 y=2 x+4 y=5

Name: Problem 16, Chapter 5: Beginning and Intermediate Algebra
Uploaded: 2020-06-26T20:39:39-08:00
Duration: 4 min 44 s
Channel: Joshua Fischbach
Description: Solve each system by substitution. 7 x+6 y=2 x+4 y=5

Solve each system by substitution. 7 x+6 y=2 x+4...

Key Concepts

Verification of Solutions

After obtaining a solution for the system of equations, verification is essential to ensure accuracy. This involves plugging the solution back into the original equations to confirm that they are all satisfied. Verification helps catch any potential mistakes made during the substitution and simplification steps.

Substitution Method

The substitution method involves solving one of the equations for one variable in terms of the others and then substituting this expression into the remaining equations. This reduces the number of variables and equations, eventually leading to a single equation that can be solved easily. The process simplifies finding the solution to the system.

System of Linear Equations

A system of linear equations consists of two or more equations involving the same set of variables, where each equation represents a straight line when graphed. The goal is to find a set of values for the variables that simultaneously satisfy all the equations, which represents the point or points of intersection of these lines.

Transcript

00:01 Here we're trying to solve the system 7x plus 6y equals 2 and x plus 4y equals 5.

00:08 Which means we're looking for the x and y order pair that's a solution to both of these lines.

00:13 In other words, a point of intersection between the two lines.

00:16 The best process for using substitution is to get either the x or y all by itself.

00:20 It doesn't matter which one.

00:21 It doesn't matter which equation.

00:23 So pick the easiest one to work with.

00:24 This is the easiest one to work with because the x is a coefficient of 1.

00:28 So it's going to make it easier to solve for x.

00:30 So that's what we're going to do.

00:32 We're going to just grab that equation, or we're going to solve for x.

00:35 So we get x plus 4y equals 5.

00:40 You want the x by itself.

00:41 So how do you undo plus 4y, minus 4y to both sides? and you get x on its own as equal to negative 4y plus 5.

00:51 So this is the value of x in terms of y.

00:55 Now we can replace, for step two, the x in the first equation, or substitute the x in the first equation, with this value of y, negative 4x plus 5, this expression.

01:09 So we now can say that 7 times the quantity, let's put this in the color -coded form for you guys, so 7 times the quantity of x, which is the expression negative 4y plus 5, that value of x, plus 6y, plus 6y, plus 6y, is equal to 2.

01:34 Distribute, we get 7 times negative 4y, which is negative 28y, 7 times positive 5, which is positive 35, and then plus 6y, and that's equal to 2.

01:49 Combine like terms, you want to get the ys together.

01:52 What's negative 28y plus 6y, that's negative 22y, plus 35 is equal to 2.

02:03 You now want to get the negative 28y, plus 25 ,000.

02:04 You now want to get the negative 22, 2 y isolated so we want to get rid of this plus 35 how do you undo plus 35 you minus 35 not to one side but to both sides of your equation or whatever you do one side you always do to the other you get negative 22 y is equal to 2 plus negative 35 which is negative 33 and then you want to undo times by negative 22 so you're going to divide by the coefficient negative 22 to both sides because you don't want the value of negative 22 y's you want the value of 1 y's and then a negative divided by a negative is positive.

02:41 And this is the same as three halves because you can divide top and bottom by 11, right? so this is positive three halves...

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