3. Explain how you can tell if an equation has no infinite solution, write an example and solve.

I don't understand how you can tell, help?

For linear equations,

- if in your solution, the variable drops out, and you end up with a true statement, there will be an infinite number of solutions.
e.g. 3x - 9 = 3(x-3)
solving you get 0 = 0, which is true, so infinite number
- if in your solution, the variable drops out, and you end up with a false statement, there will be no solution
e.g. 3x - 8 = 3(x-3)
solving you get -8 = -9, which is false, so no solution.

For higher level equations, such as quadratics, cubic, log equations, etc
let y = (the expression forming your equation) and graph it. If it does not cross the x-axis, there will be no solution.
e.g. x^4 + x + 29 = 0
let y = x^4 + x + 29

http://www.wolframalpha.com/input/?i=plot+y+%3D+x%5E4+%2B+x+%2B+29

graph does not cross the x-axis, thus, no real number solution

To determine if an equation has no infinite solution, we need to examine its coefficients or variables. Here's a step-by-step explanation:

1. Look for variables: Start by identifying the variables in the equation. Variables are represented by letters, such as x or y.

2. Check for unique solutions: If the equation has one solution, it means that it can be solved for a single value of the variable. For example, if the equation is 2x + 3 = 7, you can solve it to find x = 2.

3. Assess the coefficients: Coefficients are the numerical values multiplied by the variables. It is important to consider both the coefficients of the variables and the constant term (the number without a variable). For example, in the equation 4x - 2 = 6, the coefficient of x is 4, and the constant term is -2.

4. Calculate the coefficient ratio: Divide the coefficient of the variable by the coefficient of the constant term. If the ratio is the same for all variables, it indicates that the equation has no infinite solution. This means that changing the values of the variables will not yield more solutions.

5. Solve the equation: With the given coefficients, solve the equation as you normally would to find the unique solution. Let's consider the equation 3x + 5y = 10 as an example.

To determine if it has no infinite solution, we need to compare the coefficient ratio for x and y:

Coefficient ratio for x = coefficient of x / coefficient of constant term = 3 / 10 = 3/10
Coefficient ratio for y = coefficient of y / coefficient of constant term = 5 / 10 = 5/10

Since the coefficient ratios are different (3/10 and 5/10), it indicates that the equation has no infinite solution.

Now, let's solve the equation to find the unique solution:

3x + 5y = 10

To solve for x, we need to isolate x on one side of the equation:

3x = 10 - 5y

Divide both sides by 3 to get the value of x:

x = (10 - 5y) / 3

This equation gives you the unique solution for x, in terms of y.

Remember, to determine if an equation has no infinite solution, you need to check if the coefficient ratios are the same for all variables.