1. For each equation below, determine if the given point is a solution: (a) $$ y=x^{3}-5 x+4 $$, is $$ (2,6) $$ a solution? (b) $$ y=\frac{x+4}{x-1} $$, is $$ (3,7) $$ a solution?

Question

UpStudy AI · Accepted Answer

Let's determine whether each point satisfies its respective equation by substituting the $$ x $$- and $$ y $$-values into the equation.

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### **(a) $$ y = x^{3} - 5x + 4 $$, is $$ (2, 6) $$ a solution?**

**Step 1: Substitute $$ x = 2 $$ into the equation to find $$ y $$.**

$$
y = (2)^{3} - 5(2) + 4 = 8 - 10 + 4 = 2
$$

**Step 2: Compare the calculated $$ y $$-value with the given $$ y $$-value (6).**

$$
2 \neq 6
$$

**Conclusion:** $$ (2, 6) $$ is **not** a solution to the equation $$ y = x^{3} - 5x + 4 $$.

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### **(b) $$ y = \frac{x + 4}{x - 1} $$, is $$ (3, 7) $$ a solution?**

**Step 1: Substitute $$ x = 3 $$ into the equation to find $$ y $$.**

$$
y = \frac{3 + 4}{3 - 1} = \frac{7}{2} = 3.5
$$

**Step 2: Compare the calculated $$ y $$-value with the given $$ y $$-value (7).**

$$
3.5 \neq 7
$$

**Conclusion:** $$ (3, 7) $$ is **not** a solution to the equation $$ y = \frac{x + 4}{x - 1} $$.

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### **Final Answer:**

**Neither point is a solution.**

- (a) $$ (2, 6) $$ does not satisfy $$ y = x^{3} - 5x + 4 $$.
- (b) $$ (3, 7) $$ does not satisfy $$ y = \frac{x + 4}{x - 1} $$.
Neither point is a solution. - For $$ y = x^{3} - 5x + 4 $$, $$ (2, 6) $$ does not satisfy the equation. - For $$ y = \frac{x + 4}{x - 1} $$, $$ (3, 7) $$ does not satisfy the equation.

1. For each equation below, determine if the given point is a solution: (a) \( y=x^{3}-5 x+4 \), is \( (2,6) \) a solution? (b) \( y=\frac{x+4}{x-1} \), is \( (3,7) \) a solution?

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