I have a few problems I need you to take a look at and tell me if I arrived at the right answer.
1. I am to use the intercepts to graph the equation x+3y=6
I used the formula y=mx+b using 3 and -6 for x I got these as my ordered pairs to graph (3, 1), (-6, 4)…Is this correct? I also got these two ordered pairs (0, 2) (3, 1)
2. Multiply -2/1*(-6/1): I got 12
3. Solve 3+4x<27 I got <6
4. Solve the compound inequality: 6>-4x+5 or 9 <(or equal to)-4x+2
My answer was (-oo,-7/4]u(-1/4,oo)
5. Is (5,2) a solution to 4x-2y=-6 I said no
6. Decide if the line is parallel, perpendicular or neither
5x+4y=2
4x-5y=4 I said neither
7. Solve using elimination: 5r-3s=11
3r+5s=61 I got (7, 8)
8. Graph using slope and y intercept: y=4/3x+3 I graphed (0, 3) and (3, 7)
9. Solve using elimination
2x+3y=1
4x+6y=2
I got 0…there is no solution
Let's go through each problem one by one and check if your answers are correct:
1. Graphing using intercepts: To find the intercepts, set x or y equal to zero in the equation.
For x=0, we get: 0 + 3y = 6, which gives y = 2. So, one intercept is (0, 2).
For y=0, we get: x + 3(0) = 6, which gives x = 6. So, the other intercept is (6, 0).
It seems like you made an error in finding the intercepts. The correct intercepts are (0, 2) and (6, 0).
2. Multiplication: When multiplying fractions, multiply the numerators together and the denominators together.
In this case, (-2/1) * (-6/1) = (2 * 6) / (1 * 1) = 12 / 1 = 12. Your answer of 12 is correct.
3. Inequality solving: Solve the inequality just like you would solve an equation, but keep in mind that the inequality sign might flip if you multiply/divide by a negative number.
To solve 3 + 4x < 27, subtract 3 from both sides: 4x < 24. Then, divide both sides by 4: x < 6. Your answer of x < 6 is correct.
4. Compound inequality solving: Solve each inequality separately and then combine the solutions with "or" or "and" depending on the inequality signs.
First inequality: 6 > -4x + 5. Subtracting 5 from both sides, we get -4x < 1. Dividing by -4, remember that the inequality sign flips: x > -1/4.
Second inequality: 9 ≤ -4x + 2. Subtracting 2 from both sides, we get -4x ≤ 7. Dividing by -4, we need to flip the inequality sign again: x ≥ -7/4.
So, combining the two inequalities, the correct answer is (-7/4, ∞) or (-1/4, ∞).
5. Testing a solution: Plug the values of x and y from the given point into the equation and see if it satisfies the equation. For (5, 2):
4(5) - 2(2) = 20 - 4 = 16, which is not equal to -6. Your answer of "no" is correct.
6. Parallel, perpendicular, or neither: To determine the relationship between two lines, compare their slopes. If the slopes are equal, the lines are parallel. If the slopes are negative reciprocals of each other, the lines are perpendicular. If neither condition is met, the lines are neither parallel nor perpendicular.
The first line's slope is -5/4, and the second line's slope is -4/5. Since the slopes are not equal and not negative reciprocals of each other, your answer of "neither" is correct.
7. Solving using elimination: Multiply the equations by appropriate numbers to make the coefficients of one variable equal but with opposite signs. Then add or subtract the equations to eliminate that variable.
Multiply the first equation by 3 and the second equation by -5 to get:
15r - 9s = 33
-15r - 25s = -305
Adding these two equations gives -34s = -272. Dividing both sides by -34, we get s = 8.
Substituting s = 8 into the first equation, we have 5r - 3(8) = 11, which simplifies to 5r - 24 = 11. Adding 24 to both sides, we get 5r = 35. Dividing both sides by 5, we find r = 7.
So, the correct solution is (7, 8).
8. Graphing using slope and y-intercept: The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept. The slope is the coefficient of x, and the y-intercept is the value of y when x = 0.
In this case, the slope is 4/3 and the y-intercept is 3. You correctly graphed the points (0, 3) and (3, 7).
9. Solving using elimination: Multiply both equations by appropriate numbers to make the coefficients of one variable opposite in sign. Then add or subtract the equations to eliminate that variable.
Multiplying the first equation by 2 and the second equation by -1 gives:
4x + 6y = 2
-4x - 6y = -2
Adding these two equations eliminates the variables and gives 0 = 0.
This means the system of equations is dependent and has infinitely many solutions. Your answer of "there is no solution" is incorrect, as there are infinitely many solutions.
I hope this helps clarify your answers and understanding of the problems! Let me know if you have any further questions.