I've been trying to figure this out but I can't seem to find the correct answer...
DIRECTIONS: Find the points of intersection of the groups of the equations, and check your results analytically.
QUESTION 1:
2x - 3y = 13
5x + 3y = 1
QUESTION 2:
x^2 + y^2 = 5
x - y = 1
THANK YOU!
queston 1) add the equations to get
7x=14 that gives you x, then put x in either of the equations to solve for y.
question 2) solve for x in the second equation, substitute that into the first equation, solve for y.
To find the points of intersection between two sets of equations, you can use one of two methods: substitution or elimination. I'll explain both methods, and then we can use them to find the points of intersection for your specific questions.
Substitution Method:
1. Solve one equation for one variable in terms of the other.
2. Substitute the expression obtained in step 1 into the other equation.
3. Solve this resulting equation for the remaining variable.
4. Substitute the value obtained in step 3 into either of the original equations to solve for the other variable.
5. Check the solution by substituting the values into both original equations.
Elimination Method:
1. Multiply one or both equations by appropriate constants to make the coefficients of one variable (x or y) the same.
2. Add or subtract the resulting equations to eliminate one variable.
3. Solve the resulting equation for the remaining variable.
4. Substitute the value obtained in step 3 into either of the original equations to solve for the other variable.
5. Check the solution by substituting the values into both original equations.
Now let's solve your specific questions:
QUESTION 1:
Equation 1: 2x - 3y = 13
Equation 2: 5x + 3y = 1
Using the elimination method, we can add the two equations together to eliminate the y variable:
(2x - 3y) + (5x + 3y) = 13 + 1
7x = 14
x = 2
Substituting x = 2 into either equation:
2(2) - 3y = 13
4 - 3y = 13
-3y = 13 - 4
-3y = 9
y = -3
The point of intersection for these two equations is (2, -3).
QUESTION 2:
Equation 1: x^2 + y^2 = 5
Equation 2: x - y = 1
Using the substitution method, we solve Equation 2 for x:
x = 1 + y
Substituting into Equation 1:
(1 + y)^2 + y^2 = 5
1 + 2y + y^2 + y^2 = 5
2y^2 + 2y + 1 = 5
2y^2 + 2y - 4 = 0
Now we can solve this quadratic equation. Factoring or using the quadratic formula, we find:
(y + 2)(2y - 2) = 0
Setting each factor equal to zero, we have:
y + 2 = 0 --> y = -2
2y - 2 = 0 --> 2y = 2 --> y = 1
Substituting either value of y into Equation 2:
x - (-2) = 1 --> x + 2 = 1 --> x = -1
x - 1 = 1 --> x = 2
The points of intersection for these two equations are (-1, -2) and (2, 1).
Make sure to check the solutions by substituting the values into both original equations to ensure they satisfy both of them.
I hope this explanation helps you understand the process of finding the points of intersection!