Solve each system by substitution. 2 y-7 x=-14 4 x-y=7

Name: Problem 14, Chapter 5: Beginning and Intermediate Algebra
Uploaded: 2020-06-26T20:24:33-08:00
Duration: 3 min 45 s
Channel: Joshua Fischbach
Description: Solve each system by substitution. 2 y-7 x=-14 4 x-y=7

step 1 Okay, so we're going to solve the system by substitution. System is the two lines, 2y minus 7x e

Solve each system by substitution. 2 y-7 x=-14 4...

Key Concepts

Solution Verification

Verifying the solution is an important step to ensure that the values obtained indeed satisfy all original equations in the system. This involves substituting the solutions back into each equation to check for consistency. Such verification confirms the accuracy of the result and the validity of the method used.

Algebraic Manipulation

Algebraic manipulation encompasses the various techniques used to simplify and solve equations, such as distributing, combining like terms, and isolating variables. Mastery of these operations is essential for efficiently and accurately solving systems of equations by methods such as substitution.

System of Linear Equations

A system of linear equations consists of multiple linear equations that share the same set of unknowns. The objective is to find the values for the variables that satisfy all the equations simultaneously. This concept is central to algebra and finds applications in various areas such as physics, economics, and engineering.

Substitution Method

The substitution method is a systematic technique used to solve systems of equations. It involves solving one of the equations for one variable and then replacing that variable in the other equation(s) with its equivalent expression. This reduces the system to a single equation with one unknown, which can then be solved using standard algebraic techniques.

Transcript

00:01 Okay, so we're going to solve the system by substitution.

00:04 System is the two lines, 2y minus 7x equals negative 14, and 4x minus y equals 7.

00:10 We're looking for the ordered pair with these two lines intersect.

00:14 So the first step is to get either the x or y by itself.

00:18 And i'm looking at these two equations.

00:20 It's easier to get the y by itself in this equation than anything else because it has a coefficient of negative 1.

00:26 And whenever you have a coefficient of 1 or negative 1 on either of your terms, it's easier to solve for that particular relationship.

00:33 So that's what we're going to do.

00:34 We're going to grab that equation, 4x minus y plus 7, and we're going to solve it for y.

00:42 So if you have 4x minus y equal 7, and you want to get the y by itself, you're going to just undo the plus 4x to both sides, so minus 4x to both sides.

00:54 You're going to get negative y equals negative 4x plus 7.

00:58 And if negative y is equal to negative 4x plus 7, that means that you must have a situation where y is the opposite of that, right? so, y is going to be equal to not negative 4x but positive 4x, not positive 7, but negative 7.

01:14 So now we've got the value of y.

01:17 And once we have the y by itself in this equation, we can now substitute for that into the other equation, right? so we're going to take that value of y and replace it in the first equation that we have here.

01:27 So that's step two.

01:29 So now we take that first equation, 2y minus 7x equals negative 14.

01:33 We replaced the y with 4x minus 7, and we solve for the x value of the intersection.

01:44 Let's do that.

01:48 So 2 times 4x is going to be 8x.

01:51 2 times negative 7 is negative 14.

01:54 So what you have here is you have 8x minus 7x as a light terms that you combine, which is going to be 1 positive x.

02:02 So x minus 14 is equal to negative 14 and then you're going to want to get the x by itself...