Does this system have one solution, no solution, or infinitely many solutions? Write another system of equations with the same number of solutions that uses the first equation only... Does this system have one solution, no solution, or infinitely many solutions? Write another system of equations with the same number of solutions that uses the first equation only. 12x + 51y = 156 -8x - 34y = -104
Understand the Problem
The question asks to determine the number of solutions the given system of equations has (one, none, or infinitely many). Additionally, it asks to create another system of equations that utilizes the first equation from the given system and has the same number of solutions.
Answer
Infinitely many solutions. $$ 12x + 51y = 156 $$ $$ 24x + 102y = 312 $$
Answer for screen readers
The given system has infinitely many solutions. A new system with infinitely many solutions that uses the first equation is: $$ 12x + 51y = 156 $$ $$ 24x + 102y = 312 $$
Steps to Solve
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Simplify the equations The given system of equations is: $$ 12x + 51y = 156 $$ $$ -8x - 34y = -104 $$ We can simplify the first equation by dividing by 3: $$ 4x + 17y = 52 $$ We can simplify the second equation by dividing by -2: $$ 4x + 17y = 52 $$
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Analyze the simplified system Now we have the system: $$ 4x + 17y = 52 $$ $$ 4x + 17y = 52 $$ Since both equations are the same, they represent the same line. Therefore, the system has infinitely many solutions.
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Create a new system with infinitely many solutions We need to create a new system that uses the first equation from the original system: $$ 12x + 51y = 156 $$ and has infinitely many solutions. This means the second equation must be a multiple of the first equation. For example, multiplying the first equation by 2 gives us: $$ 24x + 102y = 312 $$ So, the new system with infinitely many solutions is: $$ 12x + 51y = 156 $$ $$ 24x + 102y = 312 $$
The given system has infinitely many solutions. A new system with infinitely many solutions that uses the first equation is: $$ 12x + 51y = 156 $$ $$ 24x + 102y = 312 $$
More Information
A system of linear equations can have one solution, no solution, or infinitely many solutions. When the equations represent the same line, the system has infinitely many solutions.
Tips
A common mistake would be to stop after simplifying the first equation and not simplifying the second, which would make it harder to see the relationship between the equations. Also, mistakes in arithmetic could lead to the wrong conclusion about the number of solutions.
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