In this video, we will compare the deconstruct method and the traditional methodfor solving a linear equation in one variable.
While the methods are different it's important to recognizethat for both methods we do perform the same operations to both sides of theequation, which results in an equivalent equation.
For review, for the deconstruct method also referred to as thestory of the variable method, step one is to constructthe story of the variable.
Step two: deconstructthe story of the variable and then step three: applythe deconstruct story to both sides of the equationto solve the equation.
Step four: check the solution.
However, some issues canarise when using this method.
If an equation is inthe form of, let's say, this equation here where we have fiveminus two x equals nine, before we apply this method, we need to write an equivalent equation with the variable term first as negative two x plus five equals nine.
There's also an issue when wehave variables on both sides before we apply this method, you would need to addan equivalent equation with the variable term onone side of the equation as shown here.
For these reasons it maybe helpful to transition to a more traditional approachto solving linear equation.
So, let's go through these steps as well.
Using the traditional method, if the equation has decimals or fractions, one option is to multiplyboth sides of the equation to clear the fractions or decimals and then step one.
But normally the first step, because this step is optional, the first step is to simplifyeach side of the equation by clearing parenthesisand combining like terms.
Then add or subtract toisolate the variable term on one side and then finally, multiply or divide toisolate the variable and solve the equation.
And again, check the solution.
So, going back to our example, let's first constructthe story of the variable that creates this equation and then we'll writethe deconstruct story.
Starting with the variable z, to construct the equation,we first multiply by two to get two z.
Then add five to the product.
Then, because you havea negative three, here, the next step is tomultiply by negative three and then finally add three to the product and the result is 30.
For the deconstruct story, we want to undo the stepsof the construction story which means we first undo the plus three by subtracting three, and then we undo multiplyingby negative three by dividing by negative three.
The next step is to undoadding five by subtracting five and then finally weundo multiplying by two by dividing by two.
Let's begin by subtractingthree on both sides of the equation.
Simplifying both sides, threeplus three is zero on the left and 30 minus three is 27 on the right.
The equation simplifiesto negative three times the quantity two z plus five equals 27.
The next step is to divideboth sides by negative three.
Simplifying: negative threedivided by negative three is one.
The equation simplifies to two z plus five equals, 27 divided by negativethree is negative nine.
Next step is to undo theplus five by subtracting five on both sides.
Simplifying: five minus five is zero.
We now have the equation two z equals negative nine minus five is negative 14.
Last step is to undo multiplyingby two by dividing by two.
Simplifying: two divided by two is one.
One times z is z.
On the right, negative 14divided by two is negative seven.
Our solution is z equals negative seven.
And now to solve this equation again using a more traditional method.
Again, because you don't haveany fractions or decimals the first step is to simplifyboth sides of the equation which means, in this case,we distribute negative three to begin.
So, negative three timestwo z is negative six z.
Negative three times five is negative 15 plus negative 15 is equivalent to minus 15 and we still have plusthree equals thirty.
We can still simplify theleft side of the equation by combining like terms.
Negative 15 plus threeis equal to negative 12.
We can rewrite the left side as negative six z plusnegative 12 or minus 12 equals 30.
The next step is to addor subtract to isolate the variable term.
The variable term is negative six z.
We want to undo the minus 12.
In order to undo minus 12,we add 12 to both sides of the equation.
Negative 12 plus 12 is zero.
We now have the equationnegative six z equals 42.
The last step is to multiplyor divide to solve for z.
Negative six z means negative six times z and therefore to undo the multiplication we divide both sides by negative six.
Negative six dividedby negative six is one.
One times z is z.
On the right, 42 divided by negativesix is negative seven.
So, of course, we do get the same solution using the two different methods.
But again it's important toremember for both methods we did perform the sameoperation to both sides of the equation each time, creating an equivalent equation.
Let's verify the solution is correct by substituting negativeseven for z in the equation to make sure it satisfies the equation.
Performing the substitution, we have, negative three times thequantity two times negative seven plus five, plus three equals thirty.
Simplifying the left side of the equation, we simplify inside the parenthesis first.
We multiply before adding.
This simplifies to negative three times the quantity negative 14 plus five plus three equals 30.
Still simplifying inside the parenthesis, negative 14 plus five is negative nine.
We have negative three timesnegative nine plus three equals 30.
Simplifying on the left.
The next step is tomultiply: negative three times negative nine is 27.
27 plus three equals 30.
27 plus three is 30.
30 equals 30 is true, verifying our solution is correct.
I hope you found this helpful.