Examine the six different equations that follow. Determine which pairs of equations have one solution, no solution, and infinite solutions. • 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+3=2x+7 7x+3=7x+3 4x+22=8x+10 5x+5=5x+4 3x+18=5x+8 3(4x+8)=4(3x+6) What do you notice about equations that are in each of these categories? Use complete sentences to answer the questions that follow.
2x+3=2x+7
This simplifies to
3 = 7
NO solutions
7x+3=7x+3
This works for any old value of x, infinite number of solutions
4x+22=8x+10
4 x = 12
x = 3
Whew,finally one solution :)
5x+5=5x+4
oh my, 5 = 4?
No solution again
3x+18=5x+8
2 x = 10
x = 5
one solid solution
3(4x+8)=4(3x+6)
12 x + 24 = 12 x + 24
any old x :)
To determine the solutions for the given equations, we need to analyze each equation one by one.
1. 2x+3=2x+7
To solve this equation, we can start by simplifying both sides:
2x+3=2x+7
Now, we can subtract 2x from both sides to isolate the constant term:
3=7
In this case, we can see that 3 will never be equal to 7, so there is no solution to this equation. We can square around this equation.
2. 7x+3=7x+3
In this equation, the terms on both sides are identical, so subtracting 7x from both sides would result in:
3=3
This equation suggests that 3 is equal to 3, which is true. Therefore, this equation has one solution. We underline this equation.
3. 4x+22=8x+10
We can start by simplifying the equation:
4x+22=8x+10
Subtracting 4x from both sides:
22=4x+10
Next, we subtract 10 from both sides:
12=4x
Dividing both sides by 4, we find:
x=3
Thus, this equation has one solution. We underline this equation.
4. 5x+5=5x+4
By subtracting 5x from both sides:
5=4
Here, we encounter a contradiction since 5 is not equal to 4. Consequently, this equation has no solution, and we square around it.
5. 3x+18=5x+8
Now we simplify the equation:
3x+18=5x+8
Subtracting 3x from both sides:
18=2x+8
Continuing, we subtract 8 from both sides:
10=2x
Dividing by 2:
x=5
As a result, this equation has one solution. We underline this equation.
6. 3(4x+8)=4(3x+6)
Expanding the equation:
12x+24=12x+24
Here, we can see that the terms on both sides cancel each other out. This equation is true regardless of the value of x. As a result, it has infinite solutions, and we circle around this equation.
To summarize:
Equations with one solution: 7x+3=7x+3 and 3x+18=5x+8 (Underlined)
Equations with no solution: 2x+3=2x+7 and 5x+5=5x+4 (Squared Around)
Equations with infinite solutions: 3(4x+8)=4(3x+6) (Circled)
The two equations with infinite solutions are:
7x+3=7x+3
3(4x+8)=4(3x+6)
The two equations with no solution are:
4x+22=8x+10
5x+5=5x+4
The two equations with one solution are:
2x+3=2x+7
3x+18=5x+8
Equations that have one solution have different coefficients or constants for the variable being solved for. Equations with no solution have coefficients or constants such that the variable disappears, resulting in an inconsistency. Equations with infinite solutions have coefficients and constants that result in the same expression on both sides of the equation.
Consider the two equations you circled, which form a system with infinite solutions.
Solve the equations.
Consider the two equations you put a square around, which form a system with no
solution. Solve the equations.