1. How many solutions does this system of equations have? y=-2x+6
y=3(x+4)
2. What is the square root of 45?
3. solve 2(x-4)^2=14
Can anyone hep me with these questions? I'm not in a hurry or anything but I asked two hours ago. Thanks!
1. since the two equations represent straight lines, and they have different slopes, they must intersect at ONE point.
So there is one solution
2. enter 45 then the square root button on your calculator, or square root key then 45, you will have to test which way your calculator works
3. Divide the eqution by 2
(x-4)^2=7
take the square root of both sides
x-4 = +-sqrt(7)
add 4 to both sides
x = 4 +-sqrt(7)
Thanks!
On #2 we are not aloud to use calculators but other problems like this that we are doing have a radical in them. Do you know how to solve this without a calculator?
1. To find the solutions to the system of equations, we can start by representing the equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. Let's rewrite the given equations in this form:
Equation 1: y = -2x + 6
Equation 2: y = 3x + 12 (expanded form)
By comparing the coefficients of x and the constant terms in both equations, we can deduce that the slopes of these two lines are different. The first equation has a slope of -2, while the second equation has a slope of 3. Since the slopes are different, the lines are not parallel, and they will intersect at a single point.
Therefore, this system of equations has one solution.
2. To find the square root of 45, you can use a calculator or estimate it manually. If you want to estimate it manually, you can use the following steps:
- Start with an initial estimate. Since 45 lies between the perfect squares 36 (6^2) and 49 (7^2), we can use their average, which is 6.5.
- Take your initial estimate (6.5) and divide the number you want to find the square root of (45) by it: 45 / 6.5 ≈ 6.923.
- Calculate the average of your initial estimate (6.5) and the result obtained in the previous step (6.923): (6.5 + 6.923) / 2 ≈ 6.712.
- Repeat the previous step a few more times, using the average obtained in each iteration as the new estimate. The more iterations you perform, the more accurate your estimate will be.
- Continue this iterative process until you reach the desired level of accuracy or until the numbers stop changing significantly.
Using this method, you would find that the square root of 45 is approximately 6.712.
3. To solve the equation 2(x - 4)^2 = 14, you can follow these steps:
- Divide both sides of the equation by 2 to isolate the squared term: (x - 4)^2 = 7.
- Take the square root of both sides to undo the squaring operation: √[(x - 4)^2] = √7.
- Remove the square root by applying its inverse operation: x - 4 = ±√7.
- Add 4 to both sides of the equation: x = 4 ± √7.
The solutions to the equation are x = 4 + √7 and x = 4 - √7.