Question

How many solutions a system of linear equations have if: Questions. 1.the equations have different slopes? 2.the equations have the same slope and different y-intercepts. 3.the equations have the same slope and same y-intercepts. Answers. A.no solutions. B.infinetly as many solutions C.two solutions. D.one solution

Answer 1

1 - D, 2 - A, 3 - B

Explanation:

1. The equations have different slopes.

then one real solution

Example:

`\left\{\begin{array}{ccc}y=2x+2\\y=3x-5\end{array}\right`

subtract both sides of the equations

`0=-x+7`

subtract x from both sides

`x=7`

substitute it to the first equation

`y=2(7)+2\\y=14+2\\y=16`

`x=7;\ y=16`

Other explanation:

If the lines have different slopes, they intersect. The intersection coordinates are the solution to this system of equations.

2. The equations have the same slope and different y-intercepts.

then no solutions

Example:

`\left\{\begin{array}{ccc}y=-2x+3\\y=-2x-2\end{array}\right`

subtract both sides of the equations

`0=0+5\\\\0=5`

It's FALSE

Conclusion: No solutions

Other explanation:

If the lines have the same slopes, they are parallel. If they have different y-intercept, they have no common points (no solutions).

3. The equations have the same slope and same y-intercepts.

infinitely many solutions

Example:

`\left\{\begin{array}{ccc}y=3x+3\\y=3x+3\end{array}\right`

add both sides of the equations

`0=0`

It's TRUE

Conclusion: infinitely many solutions

Other explanation:

If the lines have the same slope and the same y-intercepts, then the equations shows the same line. Two overlapping straight lines have infinitely many common points (infinitely many solutions).