You may use a graphing calculator to solve these problems....

You may use a graphing calculator to solve these problems.
Which of the following equations is equivalent to the equation $3 x+5=2 x+1 ?$
(A) $3 x=2 x$
(B) $3 x=2 x+4$
(C) $\frac{3}{2} x+\frac{5}{2}=x+1$
(D) $3 x+6=2 x$
(E) $3 x=2 x-4$

Key Concepts

Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable raised to the first power. Solving linear equations involves finding the value of the variable that makes the equation true, typically using operations like addition, subtraction, multiplication, and division to isolate the variable.

Properties of Equality

The properties of equality state that if you perform the same operation on both sides of an equation, the equality is maintained. This concept is crucial for solving equations as it allows for the rearrangement and simplification of equations without changing their solutions.

Equivalent Equations

Equivalent equations are different forms of an equation that have the same solution set. Through algebraic manipulation, such as combining like terms and applying the properties of equality, an equation can be transformed into an alternate form that is equivalent to the original, allowing for easier interpretation or solution.

Transcript

00:04 We're given the equation of 3x plus 5 equals 2x plus 1.

00:14 And this is actually a multiple choice question where they're giving us several different equations and they want to see which of these are equivalent.

00:22 So we can't have more than one answer.

00:24 Well, equation a is 3x equals 2x.

00:29 Equation b is 3x equals 2x plus 4 equation c is 3 halves plus 5 halves equals x plus 1 equation d is 3x plus 6 equals 2x and equation e is 3x equals 2x minus 4.

01:01 So i'm going to approach this just one equation at a time and see if algebraically, how can i try to turn my given equation into that new equation? well, for a, for 3x equal to 2x, that means i'd have to be able to get rid of the minus 5 and the minus 1 at the same time.

01:22 And if i try getting rid of the 5, i've got to subtract 5 from both sides.

01:28 That means i end up with the 3x by itself, but i get the 2x minus 4, which saved me a little bit of work because i just found e there, but it didn't get the 2x by itself.

01:45 And if i try to get the 2x by itself by adding 1, i end up with 3x plus 4 because i have to subtract 1 to get rid of the adding 1.

01:55 So a is not going to work because mathematically i can't turn this equation into equation a.

02:08 Get rid of that.

02:10 And we'll try and turn this into 3x equals 2x plus 4, which we already sort of tried to do, but we didn't get a plus 4...

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