For which value of does the equation
have infinitely many real solutions in ?
For equations with a parameter (like p) where you are asked about 'infinitely many solutions' in x, first expand and simplify both sides completely. Then, treat it as an equation in x and use algebra to cancel x-terms; if x disappears, you get a condition involving only the parameter. Solve that condition: if it produces a true statement (like a specific parameter value), that value gives infinitely many solutions; if it produces a contradiction, there are no solutions. Always check any special parameter values that make factors zero directly in the original equation.
Hints
First simplify the algebra
Distribute on the left side so that both sides of the equation are written as sums involving x and constant terms.
Think about what 'infinitely many solutions' means
For a linear equation in x to have infinitely many solutions, what must be true about the expressions on the left and right sides? How should they compare after simplifying?
Cancel terms and see what is left
After expanding, try subtracting the same x-term from both sides. You should end up with an equation that involves only p. Solve that equation.
Desmos Guide
Enter the difference of the two sides
In Desmos, type an expression like f(x) = (p+2)*(3x-4) - (3*(p+2)*x - 4). Desmos will prompt you to add a slider for p; accept it.
Use the slider to test values of p
Move the slider for p and watch the graph of . You are looking for the p-value where the graph becomes the horizontal line for all x (the expression is always zero).
Identify the p-value that makes f(x) always zero
The p-value where the graph lies exactly on for every x is the value that makes the original equation true for all x, meaning it has infinitely many solutions. Read that p-value from the slider.
Step-by-step Explanation
Distribute and simplify both sides
Start by expanding the left side.
Left side:
So the equation becomes:
Use cancellation to remove the x-terms
Subtract from both sides. This cancels the x-terms completely:
which simplifies to:
Now the equation no longer involves x; it is just a condition on p. For infinitely many x-solutions, this condition must be true.
Solve the remaining equation for p
Solve :
Divide both sides by :
Subtract 2 from both sides:
So the equation has infinitely many real solutions in x when .