System of Equations in Real Problems World
Systems of Equations play an important role in solving real life problems. A system of equations can be used to solve problems where there is a relationship between two or more unknown variables. Below are some examples of the use of System of Equations:
1. Financial Planning:
Example: Let's say, you bought a book and a pencil. A book costs Rs 5 and a pencil costs Rs 2. You have spent a total of Rs.32. You have purchased 8 items. Now, the number of books and pencils has to be determined.
Equation:
5x + 2y = 32 (where x is the number of books and y is the number of pencils)
x + y = 8
Solution: By solving these two equations together, you can find the number of books and pencils.
2. Trade and Production:
Example: A company manufactures two types of products. It takes 3 hours per unit to manufacture the first product and 5 hours to manufacture the second product. They can work a total of 50 hours and want to produce a total of 12 products. Now, determine how many of each product to make.
Equation:
3x + 5y = 50 (where x is the number of the first product and y is the number of the second product)
x + y = 12
Solution: By solving these two equations together, you can find the number of each product.
Methods of solving simultaneous linear equations in two variables
There are various methods of solving linear equations in two variables. These methods are mainly divided into two main parts: Geometric Method and Algebraic Method.
1. Geometric Method
Graphical Method:
In this method, each simple equation is graphed as a straight line. Where the two lines intersect, is the solution of the equations.
Example: The equations x + y = 5 and 2x - y = 1 are drawn on two graphs and where they intersect is their solution point.
Geometric Observations
Solving a simple linear equation in two variables with the help of graphs gives the following conditions:
a) When two straight lines intersect at a point
If two straight lines intersect at a point, the equations have only one common solution.
Example:
y = 2x + 3,
y = -x + 1
Graphing these two equations will intersect at a point, and that point is their only common solution.
b) When two straight lines coincide
If two straight lines coincide perfectly, then there is only one straight line and the equations have an infinite number of general solutions.
Example:
y = 2x + 3
2y = 4x + 6
These two equations point to the same straight line, so they have an infinite number of common solutions.
c) When two straight lines do not intersect each other and are parallel
If two straight lines do not intersect and are parallel, then the equations have no common solution.
Example:
y = 2x + 3
y = 2x - 1
Graphing these two equations will be parallel and never intersect, so they have no common solution.
2. Algebraic Method
Substitution Method
Substitution Method is an Algebraic Method for solving linear equation with two variables. This method involves first extracting the value of a variable from one equation and then substituting that value into another equation. This gives us an equation with a single variable, which is solved to determine the value of the other variable.
Steps of solving by Substitution Method:
1. First step:
First the value of a variable is extracted from an equation.
2. Second step:
That value is substituted into the other equation and a single variable equation is obtained.
3. Third Step:
The value of a variable is determined by solving an equation with a single variable.
4. Fourth Step:
The value of the other variable is determined by substituting the extracted value into the first equation.
Example:
Let's say we have two equations:
x + y = 5
x - y = 1
Solution Steps:
1. First step:
The value of a variable is extracted from the first equation:
x + y = 5
or, y = 5 - x
2. Second step:
Substituting the value y = 5 - x into the second equation:
2x - (5 - x) = 1
or, 2x - 5 + x = 1
or, 3x - 5 = 1
3. Third Step:
The value of x is found by solving the equation:
3x - 5 = 1
or, 3x = 6
or, x = 2
4. Fourth Step:
The value of y is determined by substituting the value x = 2 into the first equation:
x + y = 5
or, 2 + y = 5
or, y = 3
So, the solutions of the equations are x = 2 and y = 3
Elimination Method
Elimination Method is an Algebraic Method for solving linear equation with two variables. In this method, a single variable equation is created by eliminating one variable from two equations. This allows us to set the value of one variable, which is then used to set the value of another variable.
1. First step:
The coefficients of any one variable in two equations are equalized. For this the equations are multiplied if necessary.
2. Second step:
Adding or subtracting two equations eliminates one variable and creates a new equation with a single variable.
3. Third Step:
The value of the unit variable is determined by solving the new equation.
4. Fourth Step:
The value of the other variable is determined by substituting the determined value into one of the original equations.
Example:
Let's say we have two equations:
2x + 3y = 8
4x - y = 2
Solution Steps:
1. First step:
Equating the coefficients of one of the variables in the two equations (2x + 3y = 8) and (4x - y = 2), multiplying the first equation by 2:
2(2x + 3y) = 2 ✕ 8
or, 4x + 6y = 16
2. Second step:
By subtracting the two equations (4x + 6y = 16) and (4x - y = 2) the x variable must be eliminated:
(4x + 6y) - (4x - y) = 16 - 2
or, 4x + 6y - 4x + y = 14
or, 7y = 14
3. Third Step:
Determining the value of the y variable:
or, 7y = 14
or, y = 14 7
or, y = 2
4. Fourth Step:
Substitute the value y = 2 into either original equation to determine the value of x:
2x + 3y = 8
or, 2x + 3 ✕ 2 = 8
or, 2x + 6 = 8
or, 2x = 8 - 6
or, 2x = 2
or, x = 1
So, the solutions of the equations are x = 1 and y = 2
Cross Multiplication Method
Cross Multiplication Method is an Algebraic Method for solving linear equation with two variables. In this method the solution of the equation is determined by cross multiplying the coefficients of two simple equations.
Two linear Equations
Let us assume that two simple linear equations in two variables are given:
a1x + b1y = c1
a2x + b2y = c2
Steps of Cross Multiplication Method:
1. First step:
Cross multiplying the two equations' coefficients creates two new equations.
2. Second step:
The values of the variables x and y are determined from the two equations.
Equation set:
x(b1c2 - b2c1) = y(c1a2 - c2a1) = 1(a1b2 - a 2b1)
here,
x = b 1c2 - b2c1a1b2 - a2b1
y = c 1a2 - c2a1a1b2 - a2b1y
Example:
Let's say our two equations are:
2x + 3y = 5
4x + 6y = 7
Solution Steps:
1. First step:
Here,
a1 = 2, b1 = 3, c1 = 5
a2 = 4, b2 = 6, c2 = 7
2. Second step:
x = (3 ✕ 7 ) - (6 ✕ 5)(2 ✕ 6) - (4 ✕ 3) = 21 -30 12 -12 = - 9 0 (which is undecidable)
y = (5 ✕ 4 ) - (7 ✕ 2)(2 ✕ 6) - (4 ✕ 3) = 20 - 14 12 - 12 = 7 0 (which is undecidable)
Here the solution is indeterminate, so it is an inconsistent equation.
Consistency of Two Simultaneous Linear Equations
Consistent
We can find an unique Solution:
If two straight lines intersect at a point, the equations have a common solution.
Example:
Steps of Cross Multiplication Method:
1. First step:
Cross multiplying the two equations' coefficients creates two new equations.
2. Second step:
The values of the variables x and y are determined from the two equations.
Equation set:
x(b1c2 - b2c1) = y(c1a2 - c2a1) = 1(a1b2 - a 2b1)
here,
x = b 1c2 - b2c1a1b2 - a2b1
y = c 1a2 - c2a1a1b2 - a2b1y
Example:
Let's say our two equations are:
2x + 3y = 5
4x + 6y = 7
Solution Steps:
1. First step:
Here,
a1 = 2, b1 = 3, c1 = 5
a2 = 4, b2 = 6, c2 = 7
2. Second step:
x = (3 ✕ 7 ) - (6 ✕ 5)(2 ✕ 6) - (4 ✕ 3) = 21 -30 12 -12 = - 9 0 (which is undecidable)
y = (5 ✕ 4 ) - (7 ✕ 2)(2 ✕ 6) - (4 ✕ 3) = 20 - 14 12 - 12 = 7 0 (which is undecidable)
Here the solution is indeterminate, so it is an inconsistent equation.
Consistency of Two Simultaneous Linear Equations
The solvability of two simple systems of equations is determined by whether the equations have solutions and, if so, how many solutions there are. Based on this, equations can be divided into three categories:
Consistent
We can find an unique Solution:
If two straight lines intersect at a point, the equations have a common solution.
Example:
a1x + b1y = c1
a2x + b2 y = c2
if a1a2 ≠ b1b2, but there will be a simple solution.
Example:
2x + 3y = 5
4x - y = 1
These equations have a general solution.
Inconsistent
In this characteristics, there is no Solution:
If two straight lines are parallel and never intersect, the equations have no solutions.
Example:
1. Draw the graph of the quadratic function:
First the graph of the function y = ax2 + bx + c is drawn.
The graph of this function will be an ellipse, which will intersect the y -axis.
2. Field Determination:
The points where the graph intersects the x-axis determine the x-value.
Example:
Let's say our equation is x2 - 4x + 3 = 0
1. Quadratic Functions:
y = x2 - 4x + 3
2. Draw the graph:
The graph of this function is an ellipse.
To determine the value of y, we need to find y with different values of x and draw the graph using those values.
3. Determination of point of intersection of axes:
The points where the graph of this function intersects the x-axis are our solutions.
The graph of the function x2 - 4x + 3 = 0 will intersect the x-axis at the points x = 1 and x = 3.
Analysis:
x = 1 and x = 3 are the solutions of our quadratic equation.
Advantages of solving quadratic equations with graphs:
1. Visible solutions:
We can easily visualize the solutions through graphs.
2. Determination of nature of values:
The graph shows how many real solutions the equation has and whether they are equal or different.
Example:
2x + 3y = 5
4x - y = 1
These equations have a general solution.
Inconsistent
In this characteristics, there is no Solution:
If two straight lines are parallel and never intersect, the equations have no solutions.
Example:
a1x + b1y = c1 and
a2x + b 2y = c2
if a1a2 = b1b2 ≠ c1c2, but no solution.
Example:
2x + 3y = 5
4x + 6y = 10
These equations have no solution because they are parallel.
Infinitely Many Solutions
In this characteristics, there are infinitely many Solutions:
If two straight lines are the same, then the equations have an infinite number of solutions.
Example:
Example:
2x + 3y = 5
4x + 6y = 10
These equations have no solution because they are parallel.
Infinitely Many Solutions
In this characteristics, there are infinitely many Solutions:
If two straight lines are the same, then the equations have an infinite number of solutions.
Example:
a1x + b1y = c1
a2x + b2 y = c2
if a1a2 = b 1b2 = c 1c2, but there will be infinite solutions.
Example:
2x + 3y = 6
4x + 6y = 12
These equations have an infinite number of solutions because they point to the same line.
In summary:
Consistent:
Intersecting at a Point: A Unique Solution
Same Line: Infinitely Many Solutions
Inconsistent:
Parallel Lines: No Solution
General solution method of quadratic equation in one variable
Quadratic equation: ax2 + bx + c = 0
Quadratic equation formula -
x = -b +√ b2 - 4ac 2a
Discriminant
What does discriminant mean:
b2 - 4ac
Discriminant determines the nature of the value of the quadratic equation.
Nature of value according to Discriminant
1. If b2 - 4ac = 0
The values will be real and equal.
Two equal origins: x = - b2a
Example:
Example:
2x + 3y = 6
4x + 6y = 12
These equations have an infinite number of solutions because they point to the same line.
In summary:
Consistent:
Intersecting at a Point: A Unique Solution
Same Line: Infinitely Many Solutions
Inconsistent:
Parallel Lines: No Solution
Their solvability can be determined by comparing the coefficients of the equations. This is an important step in solving linear equations that helps to understand if there are solutions and what kind of solutions there are.
General solution method of quadratic equation in one variable
Quadratic equation: ax2 + bx + c = 0
Quadratic equation formula -
x = -b +√ b2 - 4ac 2a
Discriminant
What does discriminant mean:
b2 - 4ac
Discriminant determines the nature of the value of the quadratic equation.
Nature of value according to Discriminant
1. If b2 - 4ac = 0
The values will be real and equal.
Two equal origins: x = - b2a
Example:
x2 - 2x + 1 = 0
Here, a = 1, b = - 2, and c = 1
Discriminant = ( -2)2 - 4⋅1⋅1 = 4 - 4 = 0
Discriminant = ( -2)2 - 4⋅1⋅1 = 4 - 4 = 0
Solution: x = 22 = 1 (two equal roots: 1 and 1)
2. If b2 - 4ac＞0 and is an integer:
The values will be real, unequal and rational.
Example:
2. If b2 - 4ac＞0 and is an integer:
The values will be real, unequal and rational.
Example:
x2 - 3x + 2 = 0
Here, a = 1, b = - 3, and c = 2
Discriminant = ( -3)2 - 4⋅1⋅2 = 9 - 8 = 1 (square)
Discriminant = ( -3)2 - 4⋅1⋅2 = 9 - 8 = 1 (square)
Solution: Two real, unequal and rational roots (1 and 2)
3. If b2 - 4ac＞0 and is not an integer:
The values will be real, unequal and irrational.
Example:
3. If b2 - 4ac＞0 and is not an integer:
The values will be real, unequal and irrational.
Example:
x2 - 2x - 3 = 0
Here, a = 1, b = - 2, and c = - 3
Discriminant = ( -2)2 - 4⋅1⋅( - 3) = 4 + 12 = 16 (not square)
Discriminant = ( -2)2 - 4⋅1⋅( - 3) = 4 + 12 = 16 (not square)
Solution: Two real, unequal and irrational values.
4. If b2 - 4ac＜0:
Quadratic equations have no real value.
Example: x2 + x + 1 = 0 Here, a = 1, b = 1, and c = 1
Discriminant = 12 - 4⋅1⋅1 = 1 - 4 = -3 Solution: There is no real value.
In summary:
b2 - 4ac = 0: real and equal values.
b2 - 4ac＞0 and integers: real, unequal and rational values.
b2 - 4ac＞0 and not integers: real, unequal and irrational values.
b2 - 4ac＜0: no real value.
In this way, the nature of the values of the quadratic equation can be determined using b2 - 4ac or the discriminant.
Solving Quadratic Equations with Graphs
Steps of solving Quadratic Equations with Graphs:
4. If b2 - 4ac＜0:
Quadratic equations have no real value.
Example: x2 + x + 1 = 0 Here, a = 1, b = 1, and c = 1
Discriminant = 12 - 4⋅1⋅1 = 1 - 4 = -3 Solution: There is no real value.
In summary:
b2 - 4ac = 0: real and equal values.
b2 - 4ac＞0 and integers: real, unequal and rational values.
b2 - 4ac＞0 and not integers: real, unequal and irrational values.
b2 - 4ac＜0: no real value.
In this way, the nature of the values of the quadratic equation can be determined using b2 - 4ac or the discriminant.
Solving Quadratic Equations with Graphs
Quadratic equations are usually of the form ax2 + bx + c = 0. To solve this equation graphically, we must first graph the quadratic function. This graph is usually in the form of a parabola.
Steps of solving Quadratic Equations with Graphs:
1. Draw the graph of the quadratic function:
First the graph of the function y = ax2 + bx + c is drawn.
The graph of this function will be an ellipse, which will intersect the y -axis.
2. Field Determination:
The points where the graph intersects the x-axis determine the x-value.
Example:
Let's say our equation is x2 - 4x + 3 = 0
1. Quadratic Functions:
y = x2 - 4x + 3
2. Draw the graph:
The graph of this function is an ellipse.
To determine the value of y, we need to find y with different values of x and draw the graph using those values.
3. Determination of point of intersection of axes:
The points where the graph of this function intersects the x-axis are our solutions.
The graph of the function x2 - 4x + 3 = 0 will intersect the x-axis at the points x = 1 and x = 3.
Analysis:
x = 1 and x = 3 are the solutions of our quadratic equation.
Advantages of solving quadratic equations with graphs:
1. Visible solutions:
We can easily visualize the solutions through graphs.
2. Determination of nature of values:
The graph shows how many real solutions the equation has and whether they are equal or different.
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