solve the system on linear equations

Question

solve the systems of linear equations for the problems (i have two seperate questions in here)

1.) y= 3x+4

y-x=2

2.) y-4x=3

2y=8x+5

thank you

Answer 1

1.) y= 3x+4 y=3x + 4 Subtract equations: x= -1, y= 1

y-x=2 y = x + 2

2.) y-4x=3

2y=8x+5 divide this by 2 yields: y = 4x +5/2

Since y's equal two different answers, there is no solution to this problem.

Answer 2

Hey Sam,

One method to solving a system of equations is to take one equation, isolate x or y on one side, then plug it into the other equation. Let's try that with question 1.

Equation 1: y= 3x+4

Equation 2: y-x=2

Since equation 1 already has y isolated on one side of the equation, we can plug in 3+4x for y in equation 2:

y-x = 2

3x+4-x = 2

Now solve for x:

2x+4 = 2

2x = -2

x= -1

Now that you know x=-1, you can plug it back into either equation. It's probably easiest to plug it back in to equation 2:

y-x = 2

y- (-1) = 2

y+1 = 2

y=1

So your solution is x = -1 and y= 1. Notice that if you were to plot these lines on a graph, this is the point where they would intersect.

Now let's do the same thing for question 2:

Equation 3: y-4x=3

Equation 4: 2y=8x+5

I would isolate y in equation 3

y=3+4x

Now plug (3+4x) for y into equation 4 and solve for x:

2(3+4x) = 8x+5

6+8x= 8x +5

If you try to solve this equation, the x values end up canceling each other out so there is no variable to solve for. This means that there is no solution. There is no value of x and y that can make both equations true. If you were to plot these lines on a graph, they would be parallel, with no point of intersection.

The answer is no solution

Hope this helps!

Answer 3

When solving systems of equations, the main goal is to find similarities in order to cancel out one of the variables and work with only one variable in the equations since that is easier to work with. We can use a couple of methods to do this. One method is "solving by substitution". Usually in the question stem, if one of the equations is in the form of " y = ...." or "x = ...", then you can substitute that specific equation into the other equation.

For example, the first system is "y = 3x + 4" and "y - x = 2". To solve this, we take the first equation (y = 3x + 4) and substitute it into (y - x = 2) since the y's in both equations are the same. Therefore (3x + 4) - x = 2. Simplifying this expression yields 2x = -2, and x = -1. Then we plug the x-value back into any of the original equations. If we plug x = -1 into the equation y = 3x + 4, we get y = 1.

For the second system with "y - 4x = 3" and "2y = 8x + 5", we can use the same method. Therefore, we can rewrite the first equation into "y = 4x + 3" and substitute the new equation into the "2y = 8x + 5" equation. Therefore, 2(4x + 3) = 8x + 5. Simplifying this equation yields 8x + 6 = 8x + 5. Because these two sides are not equal to each other (if we tried to simplify we would get the result 0 = -1, which is not true), there is No Solution.

Answer 4

1.) y= 3x+4 y = 3x + 4 Subtract the equations: 2x + 2 = 0 x = -1

y-x=2 y = x + 2 y = -1 + 2 y = 1

2.) y-4x=3 y = 4x + 3

2y=8x+5 y = 4x + 5/2 dividing this equation by 2:

Then it obvious that there is no solution.