Explain why the simultaneous equations y=1/2x+2 and 2y-x:4 have an infinite number of solutions. What is diffrent about these equations compared with the equations in the first question ( the equations were y=2x+3 and 5y-10x=5)? What is similar? ( include work to prove why and how it has an infinite number of solutions)

Your second equation must be

2y -x = 4 not 2y -x:4

By dividing both sides by 2 and rearranging, one can see that it is equivalent to the first equation.

They would plot on a graph as exactly the same line, and there are an infinite number of "point" solutions along that line.

To understand why the simultaneous equations y = (1/2)x + 2 and 2y - x = 4 have an infinite number of solutions, we need to compare them with the equations in the first question, y = 2x + 3 and 5y - 10x = 5.

In the equations y = (1/2)x + 2 and 2y - x = 4, notice that the second equation can be rearranged to x = 2y - 4.

Now, let's compare the equations.

1. Different Equations:
The equations y = 2x + 3 and 5y - 10x = 5 have a unique solution because the first equation represents a straight line with a slope of 2, and the second equation has a distinct linear equation. These equations intersect at a single point, leading to only one solution.

2. Similarities:
In both sets of equations, we can compare the coefficients of x and y in the second equation. In the first set of equations, we have -10x + 5y = 5, while in the later one, we have -x + 2y = 4.

If we multiply each equation in the first set by 2, we get -20x + 10y = 10. Now we notice that the coefficients of x and y in both sets of equations are the same: -20 and 10, respectively.

Since both sets of equations have the same coefficients for x and y in the second equation, they represent the same straight line. This means they are parallel lines, and parallel lines never intersect. As a result, these equations have an infinite number of solutions.

To mathematically prove it, we can substitute the value of x from the rearranged second equation, x = 2y - 4, into the first equation, y = (1/2)(2y - 4) + 2. Solving this equation:

y = (1/2)(2y - 4) + 2
y = y - 2 + 2
y = y

Here, we get y = y, which means that for any value of y, we will have a corresponding solution for x using the second equation x = 2y - 4.

Thus, the simultaneous equations y = (1/2)x + 2 and 2y - x = 4 have an infinite number of solutions.

The simultaneous equations y=1/2x+2 and 2y-x=4 have an infinite number of solutions because they represent two equations that are actually the same line.

To understand why these equations have an infinite number of solutions, let's analyze them individually:

1. Equation 1: y=1/2x+2
This equation represents a straight line with a slope of 1/2 and a y-intercept of 2. By graphing this equation, we can visualize its behavior.

2. Equation 2: 2y-x=4
To understand this equation, we need to rewrite it in slope-intercept form. We can isolate y by adding x to both sides and then dividing both sides by 2:
2y = x + 4
y = (1/2)x + 2

Comparing the two equations, we can see that they have the same slope and y-intercept. Thus, both equations represent the same line graphically. When two equations represent the same line, there are an infinite number of solutions because every point on the line satisfies both equations.

Now, let's compare these equations with the first set of equations (y=2x+3 and 5y-10x=5):

1. Similarity:
Both sets of equations have a common slope-intercept form, allowing us to directly compare the slopes and y-intercepts. In both cases, the slopes are equal, but the y-intercepts are different.

2. Difference:
Unlike the first set of equations, where the lines had different slopes and intersected at a single point (resulting in a unique solution), the second set of equations represent the same line. This is the key difference. In the second set, there is no specific point of intersection since the equations themselves are equivalent.

To further prove that the second set has an infinite number of solutions, we can verify it algebraically:

From the first set of equations, we have:
y = 2x + 3 (Equation A)
5y - 10x = 5 (Equation B)

To make the comparison easier, let's rewrite Equation B in slope-intercept form:
5y = 10x + 5
y = 2x + 1 (Equation C)

Now, comparing Equations A and C, we can see that they are identical. Both represent the same line, which means there are an infinite number of solutions to this system of equations.

In summary, the second set of equations y=1/2x+2 and 2y-x=4 have an infinite number of solutions because they represent the same line, unlike the first set of equations. The similarity between them is that they have the same slope, while the difference is that the first set has different slopes, resulting in a single solution.