# Algebra Solutions

## Algebra Solutions

1. (8 points) 4x = 8 and 5y = 15
The 2 equations are:
Equation 1: 4x = 8
Equation 2: 5y = 15

Divide both sides of Equation 1 by 4 to get 4x/4 = 8/4
The simplified version of equation 1 is x = 2.

Divide both sides of Equation 2 by 5 to get 5y/5 = 15/5
The simplified version of equation 2 is y = 3.

Now, let us plot the above equations on a graph. The co-ordinates of the point of intersection (if any) of these 2 lines will give the solution.

Let’s consider the simplified equation 1, which is x = 2. There is no y in the equation. This means that whatever the value of the y co-ordinate on the graph, the x co-ordinate is always 2. This means that this line will pass through the points (2,1), (2,2), (2,3). In general, it will pass through any point (2,n), where n is a real number. However, we need only 2 points to define a line.

Locate the points (2,1) and (2,6) on the graph. Now, draw a line through these points and label it 4x = 8.

In a similar manner, notice that there is no x in the simplified equation 2, which is y = 3. This means that, for any value of x, the value of the y co-ordinate will be 3. As before, we need to locate only 2 points in order to draw the graph of equation 2. Assume these 2 points are (-4,3) and (6,3). Now, draw a line through these points and label it 5y = 15.

The graph is shown below (scale is 1 unit along x-axis = 1 unit along y-axis):
[pic]

Point A (2,0) is the x-intercept of 4x = 8.
Point B (0,3) is the y-intercept of 5y = 15.
Point P (2,3) is the unique point of intersection of the 2 lines. So, the solution of 4x = 8 and 5y = 15 is given by the co-ordinates of P. The solution is x=2, y=3.

2. (8 points) 0.2x + 0.4y = 1.7 and 8.3x - 6.3y = -4.3

The 2 equations are:
Equation 1: 0.2x + 0.4y = 1.7
Equation 2: 8.3x '' 6.3y =-4.3

Multiply both sides of Equation 1 by 5 to get
x + 2y = 8.5…eq.(3)

Multiply both sides of Equation 2 by 10 to get
83 x '' 63 y = -43…eq.(4)

We will use the...