Solving equation division refers to how to use the properties and algorithms of equations to simplify equations into easier forms when solving equations. The purpose of solving equation division is to eliminate the denominator in the equation, so that the equation becomes an equation containing only addition, subtraction and multiplication, or an equation containing only one denominator, so as to reduce the difficulty of the equation and facilitate the solution.
Solving equation division is an important algebraic skill, and it has a wide range of applications in mathematics and other disciplines, such as:
- In algebraEquation division can be used to solve various types of equations, such as unary linear equations, unary quadratic equations, fractional equations, radical equations, exponential equations, logarithmic equations, etc., and can also be used to solve systems of equations, inequalities, systems of inequalities, etc.
- In GeometryThe division of equations can be used to solve various geometric problems, such as solving the side length, angle, area, volume, etc. of the graph, and can also be used to prove the properties, relations, similarity, and congruence of the graph.
- In disciplines such as physics, chemistry, engineering, biology, etcThe division method of solving equations can be used to solve various physical quantities, chemical quantities, engineering quantities, biomass, etc., and can also be used to establish mathematical models, analyze data, design schemes, etc.
There are many ways to solve equation division, and you can choose the right method according to different equations and conditions. Here are some commonly used methods:
- General Division:This is a method used to solve equations with multiple denominators, and its basic idea is to make all the denominators on both sides of the equation the same denominator, and then remove the denominator to get an equation without a denominator. The general steps of this method are as follows:
Find the least common multiple of all the denominators in the equation as the denominator of the general fraction.
Convert all fractions on both sides of the equation into fractions with the least common multiple as the denominator by dividing the least common multiple by the original denominator to get a coefficient, and then multiply this coefficient by the original numerator to get the new numerator.
Remove the denominator on either side of the equation to get an equation without a denominator.
According to the general equation solving method, solve this equation without a denominator.
For example, to solve the equation $frac - frac = frac$, you can use the following steps:
Find the least common multiple of all the denominators in the equation, i.e., the least common multiple of $2$, $3$, and $6$, which is $6$, as the denominator of the common denominator.
Convert all fractions on both sides of the equation to a fraction with $6$ as the denominator by dividing $6$ by the original denominator to get a coefficient, and then multiply this coefficient by the original numerator to get the new numerator. The details are as follows:
frac - frac = \frac$$
frac\cdot \frac - frac\cdot \frac = \frac\cdot \frac$$
frac - frac = \frac$$
Remove the denominator on either side of the equation to get an equation without a denominator, i.e.:
3x - 2x = 1$$
According to the general equation solving method, solve this equation without a denominator, i.e.:
x = 1$$
The advantage of the general division method is that it is suitable for the situation that the equation contains multiple denominators, and the equation can be reduced to an equation without denominators, simplifying the calculationHowever, it has the disadvantage of finding the least common multiple of all denominators, which is sometimes cumbersome, and may introduce redundant solutions that need to be checked.
- Cross-multiplication: This is a method used to solve an equation that contains only two denominators, and its basic idea is to cross-multiply the numerator and denominator on both sides of the equation to get an equation without a denominator. The general steps of this method are as follows:
Write the equation as $frac = frac$, where $a$, $b$, $c$, $d$ are all algebraic formulas with unknowns.
Cross multiply the numerator and denominator on both sides of the equation, i.e., multiply $a$ by $d$ and multiply $b$ by $c$ to get an equation without a denominator, i.e.,
ad = bc$$
According to the general equation solving method, solve this equation without a denominator.
For example, to solve the equation $frac = frac$, you can use the following steps:
Write the equation in the form $frac = frac$, i.e.:
frac = \frac$$
where $a = x + 1$, $b = x - 1$, $c = 2$, $d = 3$.
Cross multiply the numerator and denominator on both sides of the equation, i.e., $x + 1$ by $3$, and multiply $x - 1$ by $2$ to get an equation without a denominator, i.e.
x + 1)\cdot 3 = (x - 1)\cdot 2$$
According to the general equation solving method, solve this equation without a denominator, i.e.:
3x + 3 = 2x - 2$$
x = -5$$
The advantage of cross-multiplication is that it is suitable for the case that the equation contains only two denominators, and the equation can be quickly converted into an equation without a denominator, simplifying the calculationHowever, it has the disadvantage of being only suitable for cases where the equation contains only two denominators, not when the equation contains multiple denominators, and it may introduce redundant solutions that need to be tested.
- Multiplication principle methodThis is a method used to solve an equation that contains a denominator, and its basic idea is to multiply both sides of the equation by the denominator at the same time to get an equation without a denominator. The general steps of this method are as follows:
Find out the denominator in the equation as a factor for multiplication.
Multiply both sides of the equation by this factor at the same time, paying attention to the distributive property and the operation of deparentheses.
Get an equation without a denominator and simplify merging similar terms.
According to the general equation solving method, solve this equation without a denominator.
For example, to solve the equation 3x+2=x1, you can use the following steps:
Find out the denominator in the equation, i.e., 3, as the factor of multiplication.
Multiply both sides of the equation by 3 at the same time, pay attention to the distributive property and the operation of removing brackets, and get:
3⋅3x+2=3⋅(x−1)
x+2=3x−3
To obtain an equation without a denominator, simplify and merge similar terms, and get:
2x= 5 Solve this equation without a denominator according to the general equation solving method, i.e.
The advantage of the x=25 multiplication principle method is that it is suitable for the situation that the equation contains only one denominator, and the equation can be directly converted into an equation without a denominator, simplifying the calculationHowever, it has the disadvantage that it is only applicable to the case that the equation contains only one denominator, not to the case that the equation contains multiple denominators, and it may introduce redundant solutions that need to be tested.
Summary. Solving equation division refers to how to use the properties and algorithms of equations to simplify equations into easier forms when solving equations. The purpose of solving equation division is to eliminate the denominator in the equation, so that the equation becomes an equation containing only addition, subtraction and multiplication, or an equation containing only one denominator, so as to reduce the difficulty of the equation and facilitate the solution.
Solving equation division is an important algebraic skill, it has a wide range of applications in mathematics and other disciplines, it can be used to solve various types of equations, solve various geometric problems, solve various physical quantities, build mathematical models, analyze data, design solutions, etc. There are many ways to solve equation division, and you can choose the right method according to different equations and conditions.
Commonly used methods are the general division method, the cross multiplication method, and the multiplication principle method. The general division method is a method used to solve equations with multiple denominators, which can simplify the calculation by reducing the equation into equations without denominatorsCross-multiplication is a method used to solve equations with only two denominators, which can quickly reduce the equation to an equation without a denominator, simplifying the calculationThe multiplication principle method is a method used to solve equations that contain only one denominator, which can directly convert the equation into an equation without a denominator and simplify the calculation.