Math Problem Statement
Solution
We are tasked with finding the values of and such that is the only solution to the equation .
Step-by-step Solution:
-
Substitute into the equation: Since is the solution, we plug into the equation .
Substituting : Simplifying both sides:
-
Solve for in terms of : To find the relationship between and , solve the equation: Rearranging:
-
Ensure that is the only solution: For to be the only solution, the coefficients of on both sides of the original equation must be equal. This means must equal , so the equation doesn't have multiple solutions.
Therefore, set .
-
Substitute back into the equation for :
Final Answer:
Would you like more details on any step, or have any other questions?
Here are 5 related questions to explore:
- What happens if ? Will the equation have more than one solution?
- How would the solution change if the condition was that is the only solution?
- How can you check if a given equation has exactly one solution algebraically?
- What is the general method for solving a system of linear equations like this one?
- Can we solve this problem geometrically by interpreting the lines represented by the equations?
Tip: Always check if the equation is linear in order to guarantee a single solution when applying such conditions.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Algebra
Linear Equations
Formulas
Linear equation px + c = qx + d
Theorems
-
Suitable Grade Level
Grades 7-9
Related Recommendation
Find Values for p and q for Infinitely Many Solutions in the Equation -7x + p = qx + 1
Find Values of p and q for a Unique Solution in a Linear Equation
Find p and q for Polynomials with Common Factor x - 2
Solving for p in a System of Linear Equations: p + q + r = 4 and -p + q + r = 2
Solving Linear Equation q1 = 4 * r = -2