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.
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.
5 answers
Consider the two equations you circled, which form a system with infinite solutions.
Solve the equations.
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.
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
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.
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.