Choose a representation
Expand and collect like terms before comparing coefficients. When a specified value is a solution, substitute it into the original equation before isolating the parameter.
Worked example
The equation 4(x+2)+kx=6 x+8 is true for every real x. What is k? Expand the left side and compare the coefficient of x with the right side. The constants already match; the total x coefficient must be 6. Substitute the given numbers: 6-4 = 2. The requested value is 2.
Contrast a wrong operation
In that example, 6 follows this error: Ignore the existing x coefficient on the left. The correct requested value is 2. Explain which quantity the mistaken method is actually calculating.
Connect a second form
x=4 is a solution of kx+6=42. What is k? Substitute the specified solution into the equation. Subtract the constant term and divide by the nonzero input. Substitute the given numbers: (42-6) ÷ 4 = 9. The requested value is 9.
Transfer and check
The equation 5(x+2)+kx=7 x+10 is true for every real x. What is k? Pause before checking. Answer: 2. Expand the left side and compare the coefficient of x with the right side. The constants already match; the total x coefficient must be 7. Substitute the given numbers: 7-5 = 2.
Boundaries of the method
An identity holds for every input; a single successful substitution cannot prove it. Multiplying an entire equation preserves its solutions only when the multiplier is nonzero. This lesson samples a PSAT/NMSQT objective; its examples are original and do not claim official question status or full coverage.
The equation 4(x+2)+kx=6 x+8 is true for every real x. What is k?
- Expand the left side and compare the coefficient of x with the right side.
- The constants already match; the total x coefficient must be 6.
- Substitute the given numbers: 6-4 = 2.
- The requested value is 2.
Compare the other answer choices
Ignore the existing x coefficient on the left.
Add instead of subtracting the existing coefficient.
Treat addition of coefficients as multiplication.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
Math specifications