how do i prove the graph of a linear equation is a line? i know the slope between every point should be the same but idk how to prove it with letters. the equation is y=3/2x-9/2 but it should work with any equation i guess.
easy way: it's called a linear function for a reason.
or, you can show that the slope of the function is a constant 3/2 between any two points, so the graph has no curves or bends in it.
thanks. how do you underline the text?
i dont understand why you talk about functions but i find
say the coordinate are x1 y1, x2 y2
3/2=y1-y2 / x1-x2
= 3/2x1 - 3/2x2 / x1-x2
=3/2
it is a loop. how do i prove it pls?
to underline, use html tag < u> ... </ u>
(omit the space after the "<"
There's also < b> for bold and < i> for italic.
More general tags are filtered out, but these three can help.
There are also < sub> for subscripts and < sup> for superscripts (exponents)
To prove the slope of the graph is constant, just pick any two points (x1,y1) and (x2,y2) on the graph. The slope is, as usual (y2-y1)/(x2-x1).
Now, since y = 3/2 x - 9/2, that gives you
((3/2 x2 - 9/2) - (3/2 x1 - 9/2))/(x2-x1)
= (3/2 x2 - 3/2 x1)/(x2-x1)
= 3/2 (x2-x1)/(x2-x1)
= 3/2
no matter which two points you choose. So, the slope is the same everywhere.
thank you i see it now
To prove that the graph of a linear equation is a line, you need to demonstrate that the slope between every pair of points on the graph is the same. Let's use the equation y = (3/2)x - (9/2) as an example, but keep in mind that the process can be applied to any linear equation.
1. Choose any two points on the graph. Let's say we select the points (x₁, y₁) and (x₂, y₂).
2. Substitute the x and y values of the points into the equation y = (3/2)x - (9/2). This will give you the equations y₁ = (3/2)x₁ - (9/2) and y₂ = (3/2)x₂ - (9/2).
3. Simplify the equations obtained in the previous step.
- Multiply (3/2) by x₁ and subtract (9/2) from the result to get y₁.
- Multiply (3/2) by x₂ and subtract (9/2) from the result to get y₂.
4. Calculate the slope of the line passing through the two points using the slope formula: slope = (y₂ - y₁) / (x₂ - x₁).
5. Substitute the simplified equations from step 3 into the slope formula.
- Slope = (y₂ - y₁) / (x₂ - x₁)
- Slope = ((3/2)x₂ - (9/2)) - ((3/2)x₁ - (9/2)) / (x₂ - x₁)
6. Combine like terms in the numerator.
- Slope = [(3/2)x₂ - (3/2)x₁ - (9/2) + (9/2)] / (x₂ - x₁)
- Slope = (3/2)(x₂ - x₁) / (x₂ - x₁)
- Slope = (3/2) (cancellation occurs: x₂ - x₁ in numerator and denominator)
- Slope = 3/2
Since we obtained a constant value for the slope (3/2), we have successfully proven that the graph of the linear equation y = (3/2)x - (9/2), or any linear equation with a constant slope, is indeed a line.