How many solutions? $$ y=2 x-7 $$ $$ y=9 x+3 $$

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UpStudy AI · Accepted Answer

We have two equations:
$$
y = 2x - 7
$$
$$
y = 9x + 3
$$
Since both equations equal $$ y $$, we can set them equal to each other:
$$
2x - 7 = 9x + 3
$$

**Step 1:** Solve for $$ x $$.

Subtract $$ 2x $$ from both sides:
$$
-7 = 7x + 3
$$

Subtract $$ 3 $$ from both sides:
$$
-10 = 7x
$$

Divide both sides by $$ 7 $$:
$$
x = -\frac{10}{7}
$$

**Step 2:** Substitute $$ x $$ back into one of the equations to find $$ y $$. Using $$ y = 2x - 7 $$:
$$
y = 2\left(-\frac{10}{7}\right) - 7 = -\frac{20}{7} - 7
$$
Express $$ 7 $$ with denominator $$ 7 $$:
$$
y = -\frac{20}{7} - \frac{49}{7} = -\frac{69}{7}
$$

There is exactly one solution for the system:
$$
\left(-\frac{10}{7},\ -\frac{69}{7}\right)
$$

Thus, the system has one unique solution.
The system has one unique solution: $$ \left(-\frac{10}{7},\ -\frac{69}{7}\right) $$.

How many solutions?

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