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 I will help you determine if your answers are correct.
1. Graphing using intercepts: The equation x + 3y = 6 represents a linear equation. To find the x-intercept, we set y = 0 and solve for x. Similarly, to find the y-intercept, we set x = 0 and solve for y.
Setting y = 0, we get x + 3(0) = 6, which simplifies to x = 6. Therefore, the x-intercept is (6, 0).
Setting x = 0, we get 0 + 3y = 6, which simplifies to y = 2. Therefore, the y-intercept is (0, 2).
Your ordered pairs (3, 1) and (-6, 4) do not satisfy the equation x + 3y = 6, so they are not correct. The correct intercepts are (6, 0) and (0, 2).
2. Multiplying fractions: To multiply fractions, you multiply the numerators together and the denominators together. In this case, multiply -2/1 and -6/1:
(-2/1) * (-6/1) = (2 * 6) / (1 * 1) = 12/1 = 12.
Your answer of 12 is correct.
3. Solving the inequality: To solve 3 + 4x < 27, we need to isolate x. Subtracting 3 from both sides, we have:
4x < 24.
Dividing both sides by 4, we get:
x < 6.
Your answer of x < 6 is correct.
4. Solving the compound inequality: The compound inequality is 6 > -4x + 5 or 9 ≤ -4x + 2.
First, let's solve each inequality separately:
For 6 > -4x + 5, subtracting 5 from both sides, we get:
-4x + 1 < 6.
Simplifying further, we have:
-4x < 5.
Dividing both sides by -4, we get:
x > -1.25.
For 9 ≤ -4x + 2, subtracting 2 from both sides, we get:
-4x + 7 ≤ 9.
Subtracting 7 from both sides, we have:
-4x ≤ 2.
Dividing both sides by -4, we get:
x ≥ -0.5.
Combining the solutions, we have x > -1.25 or x ≥ -0.5. Your answer of (-∞, -7/4] ∪ (-1/4, ∞) is correct.
5. Testing a solution: To determine if (5, 2) is a solution to the equation 4x - 2y = -6, we substitute the values of x and y into the equation:
4(5) - 2(2) = -6,
20 - 4 = -6,
16 = -6.
Since 16 is not equal to -6, (5, 2) is not a solution. Your answer of no is correct.
6. Determining the relationship between lines: To determine if two lines are parallel, perpendicular, or neither, we compare their slopes. The slope-intercept form of a line is y = mx + b, where m represents the slope.
For the first equation, 5x + 4y = 2, we can rewrite it in slope-intercept form:
4y = -5x + 2,
y = (-5/4)x + 1/2.
The slope of this line is -5/4.
For the second equation, 4x - 5y = 4, we can rewrite it in slope-intercept form:
-5y = -4x + 4,
y = (4/5)x - 4/5.
The slope of this line is 4/5.
Since the slopes (-5/4 and 4/5) are not equal and not negative reciprocals of each other, the lines are neither parallel nor perpendicular. Your answer of neither is correct.
7. Solving with elimination: To solve the system of equations using elimination, we want to eliminate one variable by multiplying one or both equations by suitable factors. We'll eliminate "s" in this case.
Multiplying the first equation by 5 and the second equation by 3, we get:
(5)(5r) - (5)(3s) = (5)(11),
(3)(3r) + (3)(5s) = (3)(61).
This simplifies to:
25r - 15s = 55,
9r + 15s = 183.
Adding the two equations together, we eliminate "s":
25r - 15s + 9r + 15s = 55 + 183,
34r = 238,
r = 7.
Substituting the value of r into the first equation, we can solve for "s":
5(7) - 3s = 11,
35 - 3s = 11,
-3s = 11 - 35,
-3s = -24,
s = 8.
The solution to the system of equations is (7, 8). Your answer of (7, 8) is correct.
8. Graphing using slope and y-intercept: The equation y = (4/3)x + 3 is already in slope-intercept form (y = mx + b), where the slope is 4/3 and the y-intercept is (0, 3).
Starting from the y-intercept (0, 3), we can use the slope to find additional points. The slope 4/3 tells us that for every increase of 3 in the x-direction, we have an increase of 4 in the y-direction.
Using this information, we can plot the point (3, 7), which is 3 units to the right and 4 units up from the y-intercept. Connect the two points to graph the line.
Your graphed points (0, 3) and (3, 7) are correct.
9. Solving with elimination: To solve the system of equations using elimination, we start by multiplying the first equation by 2. This way, we can eliminate "x" when we subtract the two equations.
We have:
(2)(2x) + (2)(3y) = (2)(1),
4x + 6y = 2.
And the second equation remains the same:
4x + 6y = 2.
As you can see, both equations are identical. This means they represent the same line and the system is dependent, meaning there are infinitely many solutions.
Your answer of 0 or no solution is incorrect. The correct answer is infinitely many solutions.
I hope this clears up any confusion. Let me know if you have any further questions!