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When students have mastered the linear equation solving process (Algebra 2), it is advised that they use the shortcut to solve multiple-steps linear equations. The concept of the shortcut is the following: when a term moves (transposes) from one side of an equation to the other side, its operation changes to inverse operation.
The inverse of an addition operation is a subtraction operation. The inverse of a multiplication operation is a division operation.
The shortcut is actually a very popular world wide equation solving process.
Example 1. Solve: x - 7 = 5.
x = 5 + 7. (-7 moves to the right side and becomes +7).
Example 2. Solve: x - 4 + 9 = 3.
x = 3 + 4 - 9. (Move -4 and +9 to the right side)
Example 3. Solve: 3x/7 = 13.
x = 13(7)/3. (Move 3 and 7 to the right side)
x = 91/3
ADVANTAGES OF THE SHORTCUT:
1. Avoid double writing numbers or/and letters on both sides of the equation that consumes time and that usually are causes for errors.
Example 4. Solve 5x - 4m + 5 = 2x - 3m + 6. (total 6 terms)
Solving without shortcut:
5x - 4m + 5 = 2x - 3m + 6
- 2x = - 2x
3x - 4m + 5 = - 3m + 6
+ 4m - 5 = + 4m - 5
3x = m + 1.
(3x)/3 = (m + l)/3
x = (m + 1)/3.
Solving with shortcut: Move 2x to left side and move -4m + 5 to right side.
5x - 2x = -3m + 6 + 4m - 5. (total 6 terms)
3x = m - 1
x = (m - 1)/3.
2. It simplifies the solving operations. Before proceeding, look for a "smart move" that avoids cross multiplications and distributive multiplications.
Example 5. Solve 5/7 = 11/3(x - 2) ;
Solution. Don't do cross-multiplication and distributive multiplication. Move (x - 2) to left side and move 5 and 7 to the right side. Keep 3 in place.
(x - 2) = 7(11)/3(5) = 77/15 ;
x = 2 + 77/15 = 107/15.
Example 6. Solve 3/(x - 4) = (m + 3)/(m - 1) ;
Solution. Don't do cross multiplication and distribute multiplication.
Move (x - 4) to right side. Keep 3 in place. Move (m+3) and (m - 1) to the left side. 3(m - 1)/(m + 3) = x - 4 ;
x = 4 + 3(m - 1)/(m + 3) ;
Solving this equation (without shortcut) with cross multiplication and distributive multiplications may take too much time and may commit errors.
3. Can easily check at any time the balance of the equation and the numbers of terms /coefficients. No terms /coefficients should be missing during the transpose. There should be no new terms/coefficients added to the equation before grouping like terms.
Example 7. Solve: 3x - 2a - 3b + 5c = 2x + 3 - 4a + 3b - 4x (total 9 terms)
Solution 3x + 4x - 2x = 3 - 4a + 3b + 2a + 3b - 5c (total 9 terms)
5x = 3 - 2a + 6b - 5c ;
4. The shortcut easily helps transform math and science formulas.
Example 8. Formula: 1/R = 1/r1 + 1/r2 + 1/r3. Solve for r1 in terms of other letters.
Solution. 1/r1 = 1/R - 1/r2 - 1/r3 = (r1r2 - Rr3 - Rr2)/R.r2.r3
r1 = R.r2.r3 /(r1r2 - rr2 - rr3)
Example 9. Formula: I = mne/(mR + nr) ; Solve for R in terms of others.
Solution. First move (mR + nr) to left side and move I to right side:
mR + nr = mne/I ;
mR = mne/I - nr = (mne - Inr)/I
R = n(me - Ir)/mI
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What do you call a linear equation that has no solution?
You just write, "No Solution."
What do you call the solution of a equation derived from an original equation that is not a solution of the original equation?
That's an extraneous solution. You need to check for these when algebraically solving equations, especially when you take both sides of an equation to a power.
Create your own two-step equation with All Real Numbers as the solution Show the steps to solving the equation to prove the solution All Real Numbers?
This is a question from a Florida Virtual School class, please call your teacher for help instead. Thank you.
What do you call a number that stands alone in an equation?
You can call it a "number." To distinguish it from a variable expression, you can also call it a "constant."
What would you call a set of steps for solving a mathematical problem?
Algorithm
What do you call the three systems of linear equation?
Single answer. Coincidental (same equation), No solution.
What do you call a linear equation that has no solution?
You just write, "No Solution."
What do you call finding the value of the variables that make equations true?
Solving the equation.
What do you call the solution of a equation derived from an original equation that is not a solution of the original equation?
That's an extraneous solution. You need to check for these when algebraically solving equations, especially when you take both sides of an equation to a power.
Should any other factor be accounted for when solving an equation?
Different equations call for different steps to be followed when solving them. Exponents, parenthesis, addition, subtraction, multiplication and division are all generally used.
What do you call the first second and third terms in a quadratic equation?
1st = The quadratic term. 2nd = The linear term. 3rd = The constant term.
Create your own two-step equation with All Real Numbers as the solution Show the steps to solving the equation to prove the solution All Real Numbers?
This is a question from a Florida Virtual School class, please call your teacher for help instead. Thank you.
A long distance telephone company advertised the rates of 5 minutes for 85 cents and 10 minutes for 1.10 Use a linear equation to find the cost of a 25 minute phone call?
1.85
What do you call the numerical value that gives a true statement when it is substituted for the variable in the equation?
You call that a "solution" of the equation.
What do you call a number that stands alone in an equation?
You can call it a "number." To distinguish it from a variable expression, you can also call it a "constant."
What would you call a set of steps for solving a mathematical problem?
Algorithm
What do you call the things on right hand of an equation?
The products are on the right side of an equation.
