In the system of linear equations below, is a constant.
For which value of does the system have infinitely many solutions?
For systems questions asking about infinitely many solutions, remember that this happens only when the two equations are actually the same line written in different forms. Quickly compare coefficients: find the multiplier that connects the -coefficients and constants between the equations, then require that the -coefficients follow the same multiplier. Set up a simple equation (like ) from this relationship, solve for the parameter, and match it to the answer choice. This avoids solving the system and saves significant time.
Hints
Think about what 'infinitely many solutions' means
If two linear equations have infinitely many solutions, what does that tell you about how their graphs (the lines) are related?
Compare coefficients between the two equations
Look at how the coefficient of and the constant term in the second equation relate to those in the first equation. Is there a single number that multiplies both and to get and ?
Apply the same relationship to the y-coefficient
Once you know the factor that connects the first equation's -coefficient and constant to the second's, apply that factor to the first equation's -coefficient, , and set it equal to the second equation's -coefficient.
Desmos Guide
Enter the second equation
In Desmos, type the second equation exactly as given: 6x + 4y = 24. This will graph one line.
Test each value of k in the first equation
One at a time, type the first equation with each answer choice substituted for :
- For choice A, type
3x + 1y = 12. - For choice B, type
3x + 2y = 12. - For choice C, type
3x + 4y = 12. - For choice D, type
3x + 8y = 12. Turn each on and off as needed to see clearly.
Look for overlapping lines
For each value of , compare its line to the line from 6x + 4y = 24. The correct is the one for which the two lines lie exactly on top of each other (they look like a single line), indicating infinitely many solutions to the system.
Step-by-step Explanation
Understand the condition for infinitely many solutions
For a system of two linear equations to have infinitely many solutions, the two equations must represent the same line.
That means the coefficients of , the coefficients of , and the constants must all be multiplied by the same number from one equation to the other.
Compare the x-coefficients and constants
Write the system:
Compare the first equation to the second:
- The -coefficients: and . Here, .
- The constants: and . Here, .
So the second equation is already 2 times the first equation in terms of the -coefficient and the constant. For the lines to be the same, this same factor must apply to the -coefficient.
Use the same factor on the y-coefficient and solve for k
In the first equation, the -coefficient is . In the second equation, the -coefficient is .
For the equations to be the same line, the second -coefficient must also be 2 times the first one:
Solve for :
- Divide both sides by 2: .
So the value of that makes the system have infinitely many solutions is , which corresponds to choice B.