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

3 answers

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 \(y\):

\[ 4\left(\frac{7}{4}x - \frac{9}{4}\right) = 7x - 9 \]

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.

as a decimal version

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 \(x = 0\): \[ y = 1.75(0) - 2.25 = -2.25 \quad \Rightarrow \quad (0, -2.25) \]

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

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

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.