Determine the point of intersection.

(a) y = 4x + 6
y = -x + 1

(b) 2x + 5y = 10
x = 10

Please show how to get the point of intersection. thoroughly explain how to get point of intersection.

can someone answer this?

for (a), one method:

y = 4x + 6
y = -x + 1
For the standard equation, y = mx + b,
m is the slope. If m is the same for both equations, the lines are the same or parallel. If parallel, the will not intersect. This is not the case here. The lines do intersect.

There are two equations with two unknowns.
Solve for x and y.
One method, eliminate y by substitution, giving:
4x + 6 = -x + 1
Solve for x.
Then, substitute that value of x into either equation and solve for y.

thanks

( 1 , 0 ) POI for (a)

( 10 , 2 ) POI for (b)

Checking (a) only, (1, 0) is not correct. To check your answer, substitute the values into either equation.

So, y = 4x + 6
substituting your (1,0) answer...
0 = 4(1) + 6
or
0 = 10
which is not equal

A quick check of your answer for b shows a problem.

The second equation is:
2x + 5y = 10
Substituting the answer (10,2)...
2(10) + 5(2) = 10
or
20 + 10 = 10
Not correct

To find the point of intersection, we need to solve the system of equations algebraically. Let's go through each set of equations step by step.

(a) y = 4x + 6 and y = -x + 1:

Step 1: Set the equations equal to each other:
4x + 6 = -x + 1

Step 2: Combine like terms:
4x + x = 1 - 6
5x = -5

Step 3: Solve for x:
x = -5/5
x = -1

Step 4: Substitute the x-value into either equation to find y:
y = 4(-1) + 6
y = -4 + 6
y = 2

Therefore, the point of intersection for these equations is (-1, 2).

(b) 2x + 5y = 10 and x = 10:

Step 1: Substitute the value of x in the first equation:
2(10) + 5y = 10
20 + 5y = 10

Step 2: Solve for y:
5y = 10 - 20
5y = -10
y = -10/5
y = -2

Therefore, the point of intersection for these equations is (10, -2).

To summarize the process:
1. Set the equations equal to each other.
2. Simplify the equations by combining like terms.
3. Solve for one variable to find its value.
4. Substitute the value into either equation to solve for the other variable.
5. Write the coordinates as the point of intersection.

Keep in mind that in some cases, the system of equations may not have a point of intersection or it may have infinitely many solutions.