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.
For example, in the equation 2x+3=2x+7, both sides have the same expression 2x, so subtracting 2x from both sides of the equation results in 3=7, which is not true. Therefore, this equation has no solution.
In the equation 3x+18=5x+8, the coefficients for x are different on each side of the equation. By subtracting 3x from both sides and simplifying, we get 18=2x+8. By subtracting 8 from both sides and simplifying further, we get 10=2x. Dividing both sides by 2, we find that x=5. Therefore, this equation has one solution.
In equations with infinite solutions, such as 7x+3=7x+3, both sides of the equation are exactly the same. It can be seen that any value of x will satisfy the equation. Therefore, this equation has infinite solutions.
Consider the two equations you circled, which form a system with infinite solutions.
Solve the equations.
Let's solve the system of equations:
Equation 1: 7x + 3 = 7x + 3
When we simplify the equation, we can see that both sides are identical. This means that any value of x will satisfy the equation, resulting in infinite solutions.
Equation 2: 3(4x + 8) = 4(3x + 6)
First, distribute the 3 and 4:
12x + 24 = 12x + 24
When we simplify the equation, we can see that both sides are identical. This means that any value of x will satisfy the equation, resulting in infinite solutions.
In summary, the system of equations 7x + 3 = 7x + 3 and 3(4x + 8) = 4(3x + 6) has infinitely many solutions.
Plug more than one point into the original equations and check to show that the
equations can have more than one solution. Explain your work
Let's plug in more than one point into the original equations and check if they satisfy the equations.
Equation 1: 7x + 3 = 7x + 3
Let's try x = 0:
Left side: 7(0) + 3 = 0 + 3 = 3
Right side: 7(0) + 3 = 0 + 3 = 3
Both sides are equal to 3, so x = 0 satisfies the equation.
Now let's try x = 2:
Left side: 7(2) + 3 = 14 + 3 = 17
Right side: 7(2) + 3 = 14 + 3 = 17
Again, both sides are equal to 17, so x = 2 also satisfies the equation.
Since every value of x that we plug in results in both sides of the equation being equal, we can conclude that there are infinite solutions for this equation.
Equation 2: 3(4x + 8) = 4(3x + 6)
Let's try x = 1:
Left side: 3(4(1) + 8) = 3(4+8) = 3(12) = 36
Right side: 4(3(1) + 6) = 4(3+6) = 4(9) = 36
Both sides are equal to 36, so x = 1 satisfies the equation.
Now let's try x = -5:
Left side: 3(4(-5) + 8) = 3(-20+8) = 3(-12) = -36
Right side: 4(3(-5) + 6) = 4(-15+6) = 4(-9) = -36
Again, both sides are equal to -36, so x = -5 also satisfies the equation.
Just like in the previous equation, no matter which value of x we plug in, we find that both sides of the equation are equal. Therefore, this equation also has infinite solutions.
In conclusion, by plugging in different values of x into the original equations, we see that the equations have more than one solution. This shows that both equations have infinite solutions.
To determine if the given equations have infinite solutions, no solutions, or one solution, let's solve each equation step-by-step.
1) Equation: 7x+3=7x+3
Step 1: Start by subtracting 7x from both sides to isolate the constant term.
7x - 7x + 3 = 0
Step 2: Simplify the expression on the left side.
3 = 0
Step 3: Simplify the equation.
This equation simplifies to 3 = 0, which is not true. Therefore, this equation has no solution.
2) Equation: 3(4x+8)=4(3x+6)
Step 1: Distribute the 3 on the left side and the 4 on the right side.
12x + 24 = 12x + 24
Step 2: Subtract 12x from both sides to isolate the constant term.
12x - 12x + 24 = 12x - 12x + 24
Step 3: Simplify the expression on the left side.
24 = 24
Step 4: Simplify the equation.
This equation simplifies to 24 = 24, which is true for any value of x. Therefore, this equation has infinite solutions.
3) Equation: 4x+22=8x+10
Step 1: Subtract 4x from both sides to isolate the terms with x.
4x - 4x + 22 = 8x - 4x + 10
Step 2: Simplify the expression on the left side.
22 = 4x + 10
Step 3: Subtract 10 from both sides to isolate the term with x.
22 - 10 = 4x + 10 - 10
Step 4: Simplify the expression on the left side.
12 = 4x
Step 5: Divide both sides by 4 to solve for x.
12/4 = 4x/4
3 = x
Step 6: Simplify the equation.
This equation simplifies to 3 = x, or x = 3. Therefore, this equation has one solution.
4) Equation: 5x+5=5x+4
Step 1: Subtract 5x from both sides to isolate the constant term.
5x - 5x + 5 = 5x - 5x + 4
Step 2: Simplify the expression on the left side.
5 = 4
Step 3: Simplify the equation.
This equation simplifies to 5 = 4, which is not true. Therefore, this equation has no solution.
In summary:
- The equation 7x+3=7x+3 has no solution.
- The equation 3(4x+8)=4(3x+6) has infinite solutions.
- The equation 4x+22=8x+10 has one solution (x = 3).
- The equation 5x+5=5x+4 has no solution.
To determine the number of solutions for a given equation, we can simplify the equation and analyze its components.
For the equations with infinite solutions:
1. 7x + 3 = 7x + 3
To simplify this equation, we can subtract 7x from both sides:
-7x + 7x + 3 = 7x + 7x + 3
3 = 3
Since the equation simplifies to a true statement (3 = 3), it means that any value of x will make the equation true. Therefore, this equation has infinite solutions.
2. 3(4x + 8) = 4(3x + 6)
Distribute the coefficients:
12x + 24 = 12x + 24
Rearrange the terms by subtracting 12x from both sides:
-12x + 12x + 24 = 12x - 12x + 24
24 = 24
Similar to the previous equation, this simplifies to a true statement (24 = 24), indicating that there are infinite solutions.
Now let's move on to the equations with no solutions:
1. 4x + 22 = 8x + 10
First, subtract 4x from both sides:
-4x + 4x + 22 = 8x - 4x + 10
22 = 4x + 10
Next, subtract 10 from both sides:
22 - 10 = 4x + 10 - 10
12 = 4x
Divide both sides by 4:
12/4 = 4x/4
3 = x
The variable x in this equation simplifies to 3. However, when we substitute this value back into the original equation, we encounter a contradiction. The left side (4(3) + 22 = 34) does not equal the right side (8(3) + 10 = 34), which means there is no value of x that will satisfy this equation. Hence, it has no solution.
2. 5x + 5 = 5x + 4
Subtract 5x from both sides:
-5x + 5x + 5 = 5x - 5x + 4
5 = 4
This equation simplifies to a false statement (5 ≠ 4), implying that no value of x will fulfill the equation. Therefore, it has no solution.
Finally, let's consider the equations with one solution:
1. 2x + 3 = 2x + 7
Subtract 2x from both sides:
-2x + 2x + 3 = 2x - 2x + 7
3 = 7
This simplifies to a false statement (3 ≠ 7), which indicates that no value of x will satisfy the equation. Thus, it has no solution.
2. 3x + 18 = 5x + 8
Subtract 3x from both sides:
-3x + 3x + 18 = 5x - 3x + 8
18 = 2x + 8
Subtract 8 from both sides:
18 - 8 = 2x + 8 - 8
10 = 2x
Divide by 2:
10/2 = 2x/2
5 = x
This equation simplifies to x = 5, meaning that substituting x as 5 into the equation makes it true. Therefore, it has one solution.
In summary, equations with infinite solutions have expressions that are always true, equations with no solution result in false statements, and equations with one solution have values of x that make the equation true.