Question

solve the system. 4x-6y=-26; -2×+3y=13

Answer 1

[AI-generated answer — Bot GPT 3.5]

To solve this system of equations, we can use the method of substitution or elimination.

Method 1: Substitution
Let's solve one equation for one variable and substitute it into the other equation.

From the equation -2x + 3y = 13, we can solve it for x:
-2x = -13 - 3y
x = (13+3y)/2

Now substitute this expression for x into the first equation:
4((13+3y)/2) - 6y = -26

Multiply both sides by 2 to eliminate the fraction:
4(13+3y) - 12y = -52
52 + 12y - 12y = -52
52 = -52

This equation implies that there is no solution to the system of equations. The two equations represent parallel lines that will never intersect.

Method 2: Elimination
Let's multiply the second equation by 2 and add it to the first equation to eliminate x:

4x - 6y = -26 (equation 1)
-4x + 6y = 26 (equation 2 multiplied by -2)

Adding both equations gives:
0 = 0

In this case, the variables are eliminated and we are left with a true statement 0 = 0. This means that the system is dependent and there are infinitely many solutions. The two equations represent the same line and, therefore, intersect at all points along that line in the coordinate plane.

Answer 2

[AI-generated answer — Step-by-Step Bot]

To solve the given system of equations, we'll use the method of either substitution or elimination. Let's start with the method of elimination.

Step 1: Multiply both sides of the second equation by 2 to eliminate the coefficient of x.
-2x + 3y = 13 (original second equation)
2(-2x + 3y) = 2(13)
-4x + 6y = 26 (multiplied equation)

Now, the system becomes:
4x - 6y = -26
-4x + 6y = 26

Step 2: Add the two equations together to eliminate the variable x.
(4x - 6y) + (-4x + 6y) = (-26) + (26)
0 = 0

The result obtained is 0 = 0, which means the two original equations are dependent on each other. In other words, they represent the same line, so there are infinitely many solutions to this system.

Therefore, the system of equations is consistent and dependent, and it represents infinitely many solutions.

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