Consider the pair of linear equations below.
4x+6y=12
2x+3y=6
Part A: What is the relationship, if any, between 12 and 6?
Part B: Do the two equations have one solution, no solution, or infinitely many solution? Explain.
Part C: How can you verify your answers to Part A and B by solving algebraically?
If you multiply the second equation by 2, you will get the first equation. What does that tell you?
they are linear equationa
Part A: To determine the relationship between 12 and 6, we can compare their values. In this case, we can see that 12 is twice the value of 6: 12 = 2 * 6. Therefore, 12 is twice as large as 6.
Part B: We can determine the solution(s) to these equations by solving them algebraically. Let's use the method of substitution. We'll solve the first equation for x and substitute it into the second equation.
For the first equation, 4x + 6y = 12, we can isolate x by subtracting 6y from both sides:
4x = 12 - 6y
x = (12 - 6y) / 4
x = 3 - (3/2)y
Now we'll substitute this value of x into the second equation, 2x + 3y = 6:
2(3 - (3/2)y) + 3y = 6
6 - 3y + 3y = 6
6 = 6
Notice that the equation simplifies to 6 = 6, which is always true. This means that these two equations represent the same line. The infinite solutions indicate that the equations are dependent, implying that any values of x and y that satisfy one equation will satisfy the other as well.
In conclusion, the pair of equations has infinitely many solutions.
Part C: To verify our answers algebraically, we can substitute the obtained values back into the original equations and check if they hold true.
Let's substitute x = 3 - (3/2)y into the first equation:
4(3 - (3/2)y) + 6y = 12
12 - 6y + 6y = 12
12 = 12
The equation simplifies to 12 = 12, which is true. Now let's substitute x = 3 - (3/2)y into the second equation:
2(3 - (3/2)y) + 3y = 6
6 - 3y + 3y = 6
6 = 6
Again, the equation simplifies to 6 = 6, confirming that the chosen values are correct.
Hence, by verifying the equations algebraically, we can confirm that they have infinitely many solutions and that the values obtained in Part A are accurate.