To solve a system of equations or inequalities with more than one equation, we can use a variety of methods such as substitution, elimination, or graphing. Let's look at an example:
Example: Solve the following system of equations:
2x + 3y = 8
4x - 5y = 7
Method 1: Substitution
1. Solve one equation for one variable in terms of the other.
From the first equation, we can solve for x:
2x = 8 - 3y
x = 4 - (3/2)y
2. Substitute the expression obtained in step 1 into the other equation.
Replace x in the second equation with 4 - (3/2)y:
4(4 - (3/2)y) - 5y = 7
3. Solve the resulting equation for y.
Simplify: 16 - 6y - 5y = 7
Combine like terms: 16 - 11y = 7
Subtract 16 from both sides: -11y = -9
Divide by -11: y = 9/11
4. Substitute the value of y obtained in step 3 into either equation to find the value of x.
Using the first equation:
2x + 3(9/11) = 8
2x + 27/11 = 8
Subtract 27/11 from both sides: 2x = 8 - 27/11
Find a common denominator: 2x = 88/11 - 27/11 = 61/11
Divide both sides by 2: x = 61/22
5. The solution to the system of equations is x = 61/22 and y = 9/11.
Method 2: Elimination
1. Multiply one or both equations by appropriate constants to eliminate one of the variables.
Multiply the first equation by 4 and the second equation by 2 to make the coefficients of x the same:
8x + 12y = 32
8x - 10y = 14
2. Subtract one equation from the other to eliminate the x variable.
(8x + 12y) - (8x - 10y) = 32 - 14
Simplify: 22y = 18
Divide by 22: y = 18/22 = 9/11
3. Substitute the value of y obtained in step 2 into either equation to find the value of x.
Using the first equation:
2x + 3(9/11) = 8
2x + 27/11 = 8
Subtract 27/11 from both sides: 2x = 8 - 27/11
Find a common denominator: 2x = 88/11 - 27/11 = 61/11
Divide both sides by 2: x = 61/22
4. The solution to the system of equations is x = 61/22 and y = 9/11.
Method 3: Graphing
1. Graph both equations on the same coordinate plane.
The first equation is a line with a slope of -2/3 and a y-intercept of (0,8/3).
The second equation is a line with a slope of 4/5 and a y-intercept of (0, -7/5).
2. Find the point of intersection of the two lines.
The point of intersection is the solution to the system of equations and represents the x and y values that satisfy both equations.
3. Read the coordinates of the point of intersection.
The coordinates of the point of intersection are approximately x = 61/22 and y = 9/11.
4. The solution to the system of equations is x = 61/22 and y = 9/11.
Note: There are many other methods and variations to solve systems of equations or inequalities. The choice of method depends on the given system and personal preference.
Multiple Equations or Inequalities (2)
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