Question

Solve the linear system by graphing y=x y= 4x -9

Answer 1

We need two points to draw each line.

We choose two values of x, put to the line equation and calculate the values of y.

The coordinates of the intersection are the solution.

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y = x

for x = 0 → y = 0 → (0, 0)

for x = 2 → y = 2 → (2, 2)

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y = 4x - 9

for x = 2 → y = 4(2) - 9 = 8 - 9 = -1 → (2, -1)

for x = 3 → y = 4(3) - 9 = 12 - 9 = 3 → (3, 3)

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Look at the picture.

Answer: (3, 3) → x = 3 and y = 3

Answer 2

$y=x$ or $x=y$ is very easy to graph.

Choose whatever value of $x$ and use the same for $y$ in the same coordinate. Repeat and strike a line through all the points.

Some examples of valid points would be: $(1,1)$, $(2,2)$, $(0,0)$ and $(-1,-1)$.

Then, to graph $y=4x-9$ you have to look at it in terms of the equation of a line ($y=mx+c$) and its parts.

In this case, $m=4$ and $c=-9$.

So, you choose any value for $x$ and you solve to find its $y$ counterpart.

I will start with $x=0$:

$x=0\\\\y=4(0)-9\\y=0-9\\y=-9$

Now, we have a first coordinate: $(0,-9)$.

Repeat with a different value:

$x=5\\\\y=4(5)-9\\y=20-9\\y=11$

It gives us the coordinates $(5,11)$.

Plot those into the graph and strike a line through them.

Wherever the two lines in your Cartesian Plane cross, that is the solution for both equations. It should be $3$.

You can check it by replacing $x$ and $y$ by "3" in both formulas and equating the values of $y$:

$y=x\\y=4x-9\\\\x=4x-9\\(3)=4(3)-9\\3=12-9\\3=3$

Your solution is worked out and checked to be correct.

PS: I am attaching a picture of the graph, so you can have a visual idea of what it should look like.