Systems of linear equations
Mathematics
SYSTEMS OF LINEAR EQUATIONS 1
LINEAR EQUATIONS A linear equation in two unknowns x and y is an expression of the form: ax + by = c where a, b, and c are numbers, and where a and b are not both zero. For example:
2x - y = 3
the equality is true when x = 0, y = -3 because: 2·0 - (-3) = 3. So:
( 0, -3) is a solution of 2x - y = 3
Indeed, there are infinite solutions of the equation 2x - y = 3 x = 1, y = -1 because: 2·1 - (-1) = 2 + 1 = 3 x = 2, y = 1 because: 2·2 - 1 = 3 x = -1, y = -5 because: 2·(-1) - (-5) = - 2 +5 = 3 The set of solutions of such an equation forms a straight line in the plane. Every solution for x and y is a point in the plane:
1/8
Maths Department, IES Al-qázeres
Systems of linear equations
2
Mathematics
SYTEMS OF LINEAR EQUATIONS
A 2×2 system of linear equations is a set of 2 linear equations in 2 unknowns which must be solved simultaneously (together) so that the solutions are true in both equations. Example:
{4x2x−6y2y==126 }
The solution of our example is (3, 0) because: 2·3 + 2·0 = 6 4·3 – 6·0 = 12 We can solve such a system of equations graphically. That is, we draw the graph of the 2 lines and see where the lines intersect. The intersection point gives us the solution.
•
HOW MANY SOLUTIONS CAN SYSTEMS OF LINEAR EQUATIONS HAVE?
There can be zero solutions, 1 solution or infinite solutions (each case is explained in detail below). Note: Although systems of linear can have 3 or more equations,we are going to refer to the most common case, a system with exactly 2 lines. CASE I: AN ONLY SOLUTION This is the most common situation and it involves lines that intersect exactly once. 2/8
Maths Department, IES Al-qázeres
Systems of linear equations
Mathematics
CASE 2: NO SOLUTIONS This only happens when the lines are parallel. As you can see, parallel lines are not going to ever meet. Example of a system that has no solution: • Line 1: y = 5x +13 • Line 2: y = 5x + 12
CASE 3: INFINITE SOLUTIONS This is the rarest case and only occurs when you have the same line. Consider, for instance, the two lines below (y = 2x+1 and 2y = 4x +2). These two equations are really the same line . Example of a system that has infinite solutions: • Line 1: y = 2x + 1 • Line 2: 2y = 4x + 2
3/8
Maths Department, IES Al-qázeres
Systems of linear equations
3
Mathematics
METHODS TO SOLVE A SYSTEM
3.1. SUBSTITUTION METHOD This method functions by substituting the one x-value (or y-value) with the other. We are going to explain this by using an example.
EXAMPLE Solve:
{x2x−2yy==−13 }
1. Solve one of the equations for either “x =” or “ y = “ : x 2y =−1 x = −1 − 2y
2. Replace the “x” value in the second equation and solve the new equation for “y” : 2x − y = 3 2 −1 − 2y − y = 3 − 2 − 4y − y = 3 − 5y = 3 2 − 5y = 5 y=
5 = −1 −5
3. Place this “y” value into either of the original equations in order to solve for “x” : x 2y =−1 x 2 −1 = −1 x − 2 =−1 x = 2 − 1 = 1
The solution of our system is x = 1 , y = -1.
4. Check the solution substituting (1, -1) into both original equations, and both of them must be true: = 1 − 2 =−1 TRUE {x2x−2yy==−13 211 2−−1 −1 = 2 1 = 3 TRUE }
4/8
Maths Department, IES Al-qázeres
Systems of linear equations
Mathematics
3.2. ELIMINATION METHOD To solve the simultaneous equations, make the coefficients of one of the variables the same value in both equations. Then either add the equations or subtract one equation from the other (whichever is appropriate) to form a new equation that only contains one variable. This is referred to as eliminating the variable. We are going to explain this by using an example.
EXAMPLE Solve:
{x2x−2yy==−13 }
1. We try to get the coefficients of one of the variables to be opposites so that addition will eliminate it. We can multiply the first equation by -2 (or the second equation by 2): − 2x − 4y = 2 {x 2x2y−=y −1 = 3 2x − y = 3 } 2. We add both equations and calculate the value of y: −5y = 5 y =
5 = −1 −5
3. Place this “y” value into either of the original equations in order to solve for “x” : x 2y =−1 x 2 −1 = −1 x − 2 =−1 x = 2 − 1 = 1 The solution of our system is x = 1 , y = -1. 4. Check the solution substituting (1, -1) into both original equations, and both of them must be true: = 1 − 2 =−1 TRUE {x2x−2yy==−13 211 2−−1 −1 = 2 1 = 3 TRUE }
5/8
Maths Department, IES Al-qázeres
Systems of linear equations
Mathematics
3.3. EQUALIZATION METHOD The equalization method, consists in clearing the same unknown in both equations and then equaling the obtained expressions.
EXAMPLE Solve:
{x2x−2yy==−13 }
1. If we clear up x in both expression we have:
{
x 2y = −1 x =−1 − 2y 3 y 2x − y = 3 x = 2
}
2. The important thing is to realize that the value of x has to be the same in both equations, so that necessarily:
−1 − 2y =
3 y 2
3. We solve this equation for “y” : −1 − 2y =
3 y 2 −1 − 2y = 3 y − 2 − 4y = 3 y − 2 − 3 = 4y y 2 −5 = 5 y y =
5 y = −1 −5
4. Place this “y” value into either of the original equations in order to solve for “x” : x 2y =−1 x 2 −1 = −1 x − 2 =−1 x = 2 − 1 = 1 The solution of our system is x = 1 , y = -1. 5. Check the solution substituting (1, -1) into both original equations, and both of them must be true: = 1 − 2 =−1 TRUE {x2x−2yy==−13 211 2−−1 −1 = 2 1 = 3 TRUE }
6/8
Maths Department, IES Al-qázeres
Systems of linear equations
4
Mathematics
SOLVING PROBLEMS WITH SYSTEMS OF LINEAR EQUATIONS
You must follow these 4 steps: I. Read the problem carefully and identify the unknowns “x” and “y”. II. Write the problem like a system of linear equations. III. Solve the system of linear equations. IV. Check your solution using the problem and give the answer in words.
EXAMPLE
The sum of the digits of a two-digit number is 7. When the digits are reversed, the number is increased by 27. Find the number. We'll use “x” for the “tens” digit of the original number and “y” for the “units” digit. So the number is: xy And we know:
x y =7
Just as “26” is "10 times 2, plus 6 times 1", so our original number will be: 10x 1y The new number will have the values of the digits (represented by the variables) in reverse order: yx new number:
10y 1x
And this new number is twenty-seven more than the original number: 10y 1x = 10x 1y 27
7/8
Maths Department, IES Al-qázeres
Systems of linear equations
Now we have a system that I can solve:
Mathematics
y=7 {10y 1xx=10x 1y 27 }
First we'll simplify the second equation:
{9y − 9x =x 27y= 7y − x = 3} After reordering the variables:
{ yy − xx == 73}
We add both equations:
2y = 10 y = 5
Substituting this value into 1st equation:
y x = 7 5 x =7 x =2
The number is 25
8/8
Maths Department, IES Al-qázeres