Determine The Number of Solutions for Linear Equations

To model an equation we know is true, start with only ones tiles. Put the same number of ones on each side of the equation mat. We know that 2=2, for example, so if there are two ones on each side then the equation represented is true. Now any operations we do to both sides of the equation will continue to produce true equations.

To model an equation we know is untrue, start with only ones tiles. Put a different number of ones on either side of the equation mat. We know that 2=3, for example, is untrue, so if there are two ones on one side and three ones on the other, then the equation represented is untrue. Now, any operations we do to both sides will produce an equation that is untrue.

I noticed that my true equations always had the same number of x-tiles and ones tiles on each side of the equation mat. Even when I added ones to both sides, or added x-tiles to both sides, this stayed true.

I noticed that the untrue equations always had the same number of x-tiles, but a different number of ones tiles on each side of the equation. Even when I added x-tiles to both sides, or added ones tiles to both sides, this stayed true.

No, in order for an equation to be untrue the equation has to have the same number of x-tiles on both sides, and a different number of ones tiles. Note that having no x-tiles, means there are 0 x-tiles on each side, which is still the same number of x-tiles on each side.

The distributive property is a way of multiplying expressions that have more than one term in them. Every term in the first expression gets multiplied by every term in the second expression. For example, 3(x+4)=(3)(x)+(3)(4)=3x+12.

If there is one solution, one x is left in the equation. If the simplified equation has no variables left, then there are either no solutions or infinitely many solutions.

If you simplify an equation so that there are no variables left, and the equation you are left with is true, then there are infinitely many solutions. It means no matter what x-value you substitute in, both sides of the equation are equal.

If you simplify the equation so that there are no variables left, and the equation you are left with is untrue, then there are no solutions. It means that no matter what value of x you substitute in, both sides of the equation are never equal.

If you can see that there are a different number of x’s on each side of the equation, then the equation has only one solution, and you can find it using inverse operations. For example, in the equation 3(x+3)=2x+9, use distribution on the left side to get 3x+9=2x+9. Since there are 3 x’s on the left and 2 x’s on the right, there is only one solution.