Examine the six different equations that follow. Determine which pairs of equations have one solution, no solution, and infinite solutions. On your worksheet, put a circle around the two equations that have infinite solutions. put a square around the two equations that have no solution. underline the two equations that have one solution. 2x+37x+34x+225x+53x+183(4x+8)======2x+77x+38x+105x+45x+84(3x+6) What do you notice about equations that are in each of these categories? Use complete sentences to answer the questions that follow.
To determine which pairs of equations have one solution, no solution, and infinite solutions, we need to simplify each equation and compare them.
1) 2x + 37x + 34x + 225x + 53x + 18 = 3(4x + 8)
=> 320x + 18 = 12x + 24
=> 320x - 12x = 24 - 18
=> 308x = 6x = 6
=> x = 6/308 = 3/154
2) 2x + 77x + 38x + 10 = 5x + 4
=> 117x + 10 = 5x + 4
=> 117x - 5x = 4 - 10
=> 112x = -6
=> x = -6/112 = -3/56
3) 5x + 4 = 5x + 8
Since the variables cancel out, we are left with the equation 4 = 8, which is not possible. This equation has no solution.
4) 5x + 4 = 4
=> 5x = 4 - 4
=> 5x = 0
=> x = 0/5 = 0
5) 5x + 4 = 11x + 8
=> 5x - 11x = 8 - 4
=> -6x = 4
=> x = 4/-6 = -2/3
6) 5x + 8 = 5x + 8
This equation is true for any value of x. It has infinite solutions.
Based on these calculations, we can conclude that the pair of equations with infinite solutions are equations 3 (5x + 4 = 5x + 8) and 6 (5x + 8 = 5x + 8). We should circle these on the worksheet.
The pair of equations with no solution are equations 1 (2x + 37x + 34x + 225x + 53x + 18 = 3(4x + 8)) and 5 (5x + 4 = 5x + 11x + 8). We should square these on the worksheet.
The pair of equations with one solution are equations 2 (2x + 77x + 38x + 10 = 5x + 4) and 4 (5x + 4 = 4). We should underline these on the worksheet.
In summary, equations with infinite solutions occur when all the variables cancel out and both sides of the equation are equal. Equations with no solution occur when the variables cancel out and both sides of the equation are not equal. Equations with one solution occur when there is a single value that satisfies the equation.
To determine the pairs of equations with one solution, no solution, and infinite solutions, we need to solve each equation and examine the number of solutions.
1. 2x + 37x + 34x + 225x + 53x + 183(4x + 8)
2. 2x + 77x + 38x + 105x + 45x + 8(3x + 6)
Simplifying each equation:
1. 2x + 37x + 34x + 225x + 53x + 183(4x + 8) = 0
2. 2x + 77x + 38x + 105x + 45x + 8(3x + 6) = 0
To solve these equations, we can combine like terms:
1. (2 + 37 + 34 + 225 + 53)x + 183(4x + 8) = 0
(351x + 183)(4x + 8) = 0
2. (2 + 77 + 38 + 105 + 45)x + 8(3x + 6) = 0
(267x + 8)(3x + 6) = 0
Now, we can analyze the solutions for each case.
Equations with one solution: The equations (351x + 183)(4x + 8) = 0 and (267x + 8)(3x + 6) = 0 have one solution each because when we expand the given expressions, we obtain a quadratic equation for each term. Quadratic equations typically have two solutions, but in this case, the expression multiplied by x is always positive, which means we have one solution.
Equations with no solution: The equations (351x + 183)(4x + 8) = 0 and (267x + 8)(3x + 6) = 0 do not fall under this category because they both have one solution.
Equations with infinite solutions: None of the given equations have infinite solutions as stated.
To summarize, we notice that equations with one solution have a quadratic expression multiplied by x, equations with no solution do not exist in the given equations, and equations with infinite solutions are not present.