Understanding Equations

• An equation is a proposition indicating the equality of two algebraic expressions.
• Equations involve one or more unknowns and an equal sign.
• Examples:
  • x+1=3
  • 2x-4=7-x
  • y/(y-2) = 5
  • 2x-y=8
• The letters "x" and "y" are the unknowns.
• The set that satisfies the equality is called the solution set of that equation.
• The elements of the solution are called roots of the equation.
• Solving an equation means finding its solution set.
• Equivalent equations are two equations that have the same solution set.
• X-1=0 and 1-x=0 are equivalent because both have the same solution set, x=1.

Operations That Guarantee Equation Equivalence:

• Adding or subtracting the same polynomial on both sides of an equation maintains equivalence if the polynomial has the same variable as the equation.
  • Example:
    • From 3x=5-6x, adding 6x to both sides gives 3x+6x=5-6x+6x, simplifying to 9x=5.
• Multiplying or dividing both sides of an equation by the same constant guarantees equivalence, except when multiplying or dividing by zero.
  • Example:
    • Starting with 9x=5, dividing both sides by 9 yields (9x)/9 = 5/9, simplifying to x=5/9

Other Operations That Guarantee Equation Equivalence:

• Replacing any member of an equation with an equal expression guarantees the equivalence of equations.
  • Example:
    • x(x+2)=3 is equivalent to x² + 2x = 3.
• Applying the above operations guarantees that equality is maintained and the resulting equation is equivalent to the original.

Operations That Do Not Necessarily Produce Equivalent Equations:

• Multiplying both sides of an equation by an expression containing the variable.
  • Example:
    • Starting with x=1, multiplying both sides by x gives x²=x, rearranging to x²-x=0, and factoring to (x)(x-1)=0.
    • This implies x can be zero or one, but the original equation only allows x to be one, making the expressions not equivalent.
• Dividing both sides of an equation by an expression containing the variable.
  • Example:
    • Starting with (x-3)(x-4)=0, the roots are x=3 and x=4.
    • Dividing both sides by (x-4) gives x-3=0, which simplifies to x=3. -The root x=4 is lost, so the expressions are not equivalent.
• Elevating both sides of an equation to equal exponents often do not produce equivalant equations.
  • Example:
    • Starting with x=2, squaring both sides gives x²=4, which is rearranged to x²-4=0.
    • Factoring gives (x + 2)(x – 2) = 0.
    • In this case x=-2 is not a solutions of the original equation so the expressions are not equivalent

Notes on Equations:

• Operations 1 and 3 can produce equations with a greater number of roots, thus one should prove the roots in the original equation.
• Operation 2 can produce equations with fewer roots.

Linear Equations:

• A linear equation or first-degree equation in the variable “x” is an equation that can be written in the form 7x-10=2x. - 7x-10+(-2x)=2x+(-2x) , adding -2x to both sides. - 5x-10=0, reducing like terms. - 5x-10+10=0+10, adding 10 to both sides. - 5x=10 additive inverse and identity. - 5x/5 = 10/5, dividing both sides by 5. - x=2, simplifying.
• Solution set {2} is the result of the simplification.
• Transposition refers to the process of moving a term from one member of the equation to another. At the same time, its sign is changed.
• An equation is a linear equation if the original equation can be put in the form 5x+(-10)=0.

Solving Linear Equations:

• Example where: (7x+3)/2 - (9x-8)/4 = 6
  • Apply (4*((7x+3)/2 -(9x-8)/4 )= 6*4)
  • (4*(7x+3)/2) - 4*(9x-8)/4) = 6 * 4 distributive property
  • The expression simplifies to 2(7x+3)-(9x-8)=24
  • 14x+6-9x+8=24 distributive.
  • then it simplifies to 5x+14=24, adding similar terms.
  • 5x=10, subtracting 14 on both sides.
  • X=2 (Dividing between 5 in both members.)
  • With the final solution being {2}

Equation Roots

• The examples above only have root.
• This is a characteristic of all linear equations.
• This is written as ax+b=0 with a≠ 0, where the single root is x= -b/a
• As a class exercise, proving that the expression is equal to the equation ax=b=0

Solving Formula Varaibles:

• Variable clearing is a procedure similar to finding the range of its variable.
• For example, Clearing d of the equation
• v² = v² + 2ad
• 2ad = v² – v
• d = (v² – v²)/2a
• The equation S=P+Prt is equal to the formula for the value of S for a financial
• At an annual average rate if standard interest is r, during t years when opening p we apply:
• S = P + Prt
• S = P(1 + rt) distributive property
• S/(1+rt) = P(1+rt)/(1+rt) , Dividing between 1+rt≠ 0
• s/ (1 + rt)=P

Linear Equations contruction:

• Sometimes when trying to solve an equation that isn't leading causes one.
  • (2x + 1)² = 4(x² - 1) + (x – 1)
  • 4x² + 4x + 1 = 4x² - 4 +x-1
  • 4x − x = −5 −1, cancelling 4x2 by transposing
  • 3x = -6, simplifying
  • x = −2, dividing between 3 both members
  • With the solution set = {-2}
• Equation solution is applied to:
  • 3/(x-2) = 4/(x-3)
  • (x − 2)(x − 3)[3/(x − 2) = 4/x-3]
  • (x − 3)3 = (x – 2)4
  • Simplifying the terms with x-2, x-3, results to 3x *9=4x-8
  • *9 + 8 = 4x - 3x, transposing the terms
  • Which outputs -1=x, the solution set is {-1}
• Resolution of the equation √x² + 36 - 2 = x clears radical from there
  • √x² + 36 = x + 2
  • Clearing the radical equation in both sections result in
  • x² + 36 = (x + 2)²
  • This becomes x² + 36 = x² + 4x + 4, developing the binomial
  • 36 = 4x + 4, simplifying
  • 4x = 36 – 4, simplifying
  • 4x = 32
  • Which outputs =8, the simplified set is {-8}
• Someimes increasing power to the root it will become another and it will be done
  • Solving the equation √x + 7 + 7 = √x
  • √x + 7 = √x - 7, transposing members/Terms.
  • x + 7 = x - 14√x + 49, rising to the square of both sections.
  • 14√x = 49 - 7, despejamos √x.
  • simplifying will show us: 14√x = 42, elavtion to the square of the equation.
  • To the effect that =9, The solutin is shown that {Ø}, x=9 no es solución.

Term Transposition:

• Term Transposition is applied:
  • 5x-2 = 3x-3
• If the equation in one place and the sign change the process to change one side will happen:
  • From postitve to nagtive and from nagative to psotive in the scenario being summarized or resting:
    • 5x-3x=-3+2,

Multiplications By Coeficients

• applied to:
  • 5x-2/3 - 2x-3/2
  • (6*(5x-2)/3) - ( 6 * (2x-3)/2
  • Multipliying 6 to dissapear fractures(being the productor of 3 and 2)
  • 2 * (5x+2) = 3 * (2x-3)
  • 10x+4 = 6x-9
  • 10x-6x = -9-4
  • 3x=-13
  • x=-13/3
• to the effect of of the solution.
  • to not affect the equation bot sides need multiplying.

Equation Division:

• equation is aplied to:
  • 5x-2 = 3x-3
  • 5x3x = -3+2
  • to show 2x=-1
  • Which translates to 1/2 * 2x = -1 * 1/2
  • Thus the output equation.
  • x= -1/2;
  • is the answer of being isolated by the X.

Permitable Actions:

• Equations that
  • Summing y substacting coeficents y sumar o restar polinomios (a+b, a+x-y, x³ + 2x² + x + 7.)

    • A monomio: Is the equation: 3ª, -5b, ;

    • that has term

    • equation: a+b, x-y, ; that show 2 terms

    • To show 3: : a+b+c, x2 – 5x + 6,.