Points Available: 5
Question 1:
A straight line passes through and has a gradient of 2 . The equation of the line is:
A

Ask by Ramirez Ruiz. in Australia

Jan 17,2025

Question

Points Available: 5 Question 1: A straight line passes through $$ (1,5) $$ and has a gradient of 2 . The equation of the line is: A $$ y=2 x-3 $$

UpStudy AI · Accepted Answer

To determine the correct equation of the straight line that passes through the point $$ (1, 5) $$ with a gradient (slope) of 2, let's analyze the given option and derive the correct equation.

### Given:
- **Point**: $$ (1, 5) $$
- **Gradient (Slope)**: $$ m = 2 $$

### Option A:
$$ y = 2x - 3 $$

#### Checking Option A:
Let's verify if the point $$ (1, 5) $$ lies on the line $$ y = 2x - 3 $$:

1. Substitute $$ x = 1 $$ into the equation:
   $$ y = 2(1) - 3 $$
   $$ y = 2 - 3 $$
   $$ y = -1 $$

2. The calculated $$ y $$-value is $$-1$$, but the point has $$ y = 5 $$. Therefore, the point $$ (1, 5) $$ **does not** lie on the line $$ y = 2x - 3 $$.

**Conclusion**: **Option A is incorrect.**

### Deriving the Correct Equation:
To find the correct equation of the line, we'll use the **point-slope form** of a linear equation:

$$ y - y_1 = m(x - x_1) $$

Where:
- $$ m $$ is the slope.
- $$ (x_1, y_1) $$ is the given point.

1. Plug in the known values:
   $$ y - 5 = 2(x - 1) $$

2. Distribute and simplify:
   $$ y - 5 = 2x - 2 $$
   $$ y = 2x - 2 + 5 $$
   $$ y = 2x + 3 $$

### **Correct Equation**:
$$ \boxed{y = 2x + 3} $$

This line has a gradient of 2 and correctly passes through the point $$ (1, 5) $$.
The correct equation of the line is $$ y = 2x + 3 $$.

Points Available: 5
Question 1:
A straight line passes through and has a gradient of 2 . The equation of the line is:
A

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Points Available: 5 Question 1: A straight line passes through and has a gradient of 2 . The equation of the line is: A