What is the solution of: (7/2) x - (9/2) = 2y and 4y = 7x - 9

Answer 1

To solve the system of equations:

1. $\frac{7}{2}x-\frac{9}{2}=2y$
2. $4y=7x-9$

First, we can rearrange the first equation to express $y$ in terms of $x$:

$$\frac{7}{2}x-\frac{9}{2}=2y$$

Dividing both sides by 2:

$$\frac{7}{4}x-\frac{9}{4}=y$$

Now we can substitute this expression for $y$ into the second equation:

$$4y=7x-9$$

Substituting $y$:

$$4(\frac{7}{4}x-\frac{9}{4})=7x-9$$

Distributing the 4:

$$7x-9=7x-9$$

This equation simplifies to:

$$0=0$$

This simplification indicates that the two equations are dependent, meaning there are infinitely many solutions along a line. This means any point $(x,y)$ that satisfies one equation will also satisfy the other.

To express the solutions, we can take the form of $y$ from the first equation:

$$y=\frac{7}{4}x-\frac{9}{4}$$

This gives us a line in slope-intercept form, indicating the relationship between $x$ and $y$. Thus the solution set contains infinitely many points along this line.

Answer 2

as a decimal version

Answer 3

To express the solution in decimal form, we can convert the fractions in the equation $y=\frac{7}{4}x-\frac{9}{4}$ to decimals.

1. Convert $\frac{7}{4}$ to decimal: $$\frac{7}{4}=1.75$$

2. Convert $\frac{9}{4}$ to decimal: $$\frac{9}{4}=2.25$$

Now we can rewrite the equation for $y$ in decimal form:

$$y=1.75x-2.25$$

This equation represents the relationship between $x$ and $y$ in decimal form. Therefore, the solution set consists of all points $(x,y)$ that lie on the line defined by this equation.

For example:

• If $x=0$: $$y=1.75(0)-2.25=-2.25\Rightarrow (0,-2.25)$$

• If $x=1$: $$y=1.75(1)-2.25=1.75-2.25=-0.5\Rightarrow (1,-0.5)$$

• If $x=2$: $$y=1.75(2)-2.25=3.5-2.25=1.25\Rightarrow (2,1.25)$$

Thus, the decimal version of the solutions can be expressed with any value of $x$ via the equation $y=1.75x-2.25$. This means there are infinitely many solutions along the line represented by this equation.