Find the equation of the line perpendicular to 2y =3ᵡ -7, passing through the intersection of 2ᵡ

This is two problems. First, given: 2x + 3y + 1 = 0 and x + 4y – 1 = 0

The first thing I notice is that you have found some goofy character for x. You can not stand speling mistrakes in math. X and ᵡ are not the same as x. A math program will not tell you that you made a mistake, it will just give goofy results.

Second, another spelling problem: + and = are the same key, so you have to watch out for that.

You have two lines. Where they cross, the values of x and y satisfy both equations simultaneously, so it is called a simultaneous system. There are two ways to find that point. Substitution is when you solve one equation for one variable and substitute that into the other equation.

x + 4y – 1 = 0 The rule is you can do any valid operation on both sides of an equation and it will still be equal. Subtract (4y - 1).

x = -4y + 1 Now substitute this for x in the other equation.

2( -4y + 1) + 3y + 1 = 0 Evaluate.

-8y + 2 + 3y + 1 = 0

-5y = -3

y = 3/5 Plug this into either original equation to find x.

x = -7/5

Check your work: www.wolframalpha.com/input/?i=solve+2x+%2B+3y+%2B+1+%3D+0,+x+%2B+4y+%E2%80%93+1+%3D+0

Elimination is when you add the equations in a way to eliminate one variable. Otherwise the process is the same.

x + 4y – 1 = 0 Multiply everything by -2.

-2x - 8y + 2 = 0
2x + 3y + 1 = 0
------------------- ADD
0 - 5y + 3 = 0
y = 3/5 <-- SAME ANSWER
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Now the second problem.

2y =3x -7 Rewrite this in slope-intercept form, y = mx + b.

y = 3x/2 - 7/2 Slope = 3/2 so the perpendicular is -2/3.

Slope is rise over run. Run is horizontal distance, left to right. Run is always positive because we always go left to right. Rise is the vertical change in that same distance. A negative rise means it drops.

The equation of a line is y = mx where m is the slope. That line passes through the origin. If you want it to pass through some point (a, b) you subtract the coordinates like this: y - b = m(x - a). That is the point-slope form, and you can rewrite it in other forms if it is convenient. You may do this with any point on the line. They all reduce to the same equation.

Slope = -2/3 and the given point is (-7/5, 3/5) so that's the way we write it.

y- 3/5 = (-7/5)(x + 7/5) <-- ANSWER

There are several ways the answer could have been written, but no reason to do any more work on this.