Answer :
To determine which of the given equations represents the second street that runs parallel to the first street and passes through the point (1, 5), follow these steps:
1. Understand Parallelism:
- Two lines are parallel if they have the same slope.
2. Convert Each Given Equation to Slope-Intercept Form:
- Standard form of a line is [tex]\(ax + by = c\)[/tex].
- Slope-intercept form of a line is [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
Let's convert each given equation to determine their slopes:
- [tex]\(x + y = 6\)[/tex]:
[tex]\[ y = -x + 6 \][/tex]
Slope ([tex]\(m_1\)[/tex]) = [tex]\(-1\)[/tex]
- [tex]\(2x + y = 7\)[/tex]:
[tex]\[ y = -2x + 7 \][/tex]
Slope ([tex]\(m_2\)[/tex]) = [tex]\(-2\)[/tex]
- [tex]\(x - y = 6\)[/tex]:
[tex]\[ -y = -x + 6 \implies y = x - 6 \][/tex]
Slope ([tex]\(m_3\)[/tex]) = [tex]\(1\)[/tex]
- [tex]\(2x - y = 7\)[/tex]:
[tex]\[ -y = -2x + 7 \implies y = 2x - 7 \][/tex]
Slope ([tex]\(m_4\)[/tex]) = [tex]\(2\)[/tex]
3. Identify the Parallel Lines:
- Since the second street runs parallel to the first street, they must have the same slope.
- Assume the first street has a particular slope [tex]\(m\)[/tex]. We must match this slope with one of the given options to ensure parallelism.
4. Check Point Consistency:
- The second street must pass through the point [tex]\((1, 5)\)[/tex].
- Substitute [tex]\((1, 5)\)[/tex] into each equation to see which one is satisfied.
Let's check each equation with point [tex]\((1, 5)\)[/tex]:
- [tex]\(x + y = 6\)[/tex]:
[tex]\[ 1 + 5 = 6 \implies 6 = 6 \quad \text{(True)} \][/tex]
- [tex]\(2x + y = 7\)[/tex]:
[tex]\[ 2(1) + 5 = 7 \implies 2 + 5 = 7 \quad \text{(True)} \][/tex]
- [tex]\(x - y = 6\)[/tex]:
[tex]\[ 1 - 5 = 6 \implies -4 = 6 \quad \text{(False)} \][/tex]
- [tex]\(2x - y = 7\)[/tex]:
[tex]\[ 2(1) - 5 = 7 \implies 2 - 5 = -3 \quad \text{(False)} \][/tex]
Based on the above checks, both equations [tex]\(x + y = 6\)[/tex] and [tex]\(2x + y = 7\)[/tex] pass through the point (1, 5). However, generally, we are asked for one specific line. Assuming the parallelism condition with considering only these specific equations alone, the matched solution is first in sequence fixing our problem parameters, and hence:
The equation representing the location of the second street is:
[tex]\[ x + y = 6 \][/tex]
1. Understand Parallelism:
- Two lines are parallel if they have the same slope.
2. Convert Each Given Equation to Slope-Intercept Form:
- Standard form of a line is [tex]\(ax + by = c\)[/tex].
- Slope-intercept form of a line is [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
Let's convert each given equation to determine their slopes:
- [tex]\(x + y = 6\)[/tex]:
[tex]\[ y = -x + 6 \][/tex]
Slope ([tex]\(m_1\)[/tex]) = [tex]\(-1\)[/tex]
- [tex]\(2x + y = 7\)[/tex]:
[tex]\[ y = -2x + 7 \][/tex]
Slope ([tex]\(m_2\)[/tex]) = [tex]\(-2\)[/tex]
- [tex]\(x - y = 6\)[/tex]:
[tex]\[ -y = -x + 6 \implies y = x - 6 \][/tex]
Slope ([tex]\(m_3\)[/tex]) = [tex]\(1\)[/tex]
- [tex]\(2x - y = 7\)[/tex]:
[tex]\[ -y = -2x + 7 \implies y = 2x - 7 \][/tex]
Slope ([tex]\(m_4\)[/tex]) = [tex]\(2\)[/tex]
3. Identify the Parallel Lines:
- Since the second street runs parallel to the first street, they must have the same slope.
- Assume the first street has a particular slope [tex]\(m\)[/tex]. We must match this slope with one of the given options to ensure parallelism.
4. Check Point Consistency:
- The second street must pass through the point [tex]\((1, 5)\)[/tex].
- Substitute [tex]\((1, 5)\)[/tex] into each equation to see which one is satisfied.
Let's check each equation with point [tex]\((1, 5)\)[/tex]:
- [tex]\(x + y = 6\)[/tex]:
[tex]\[ 1 + 5 = 6 \implies 6 = 6 \quad \text{(True)} \][/tex]
- [tex]\(2x + y = 7\)[/tex]:
[tex]\[ 2(1) + 5 = 7 \implies 2 + 5 = 7 \quad \text{(True)} \][/tex]
- [tex]\(x - y = 6\)[/tex]:
[tex]\[ 1 - 5 = 6 \implies -4 = 6 \quad \text{(False)} \][/tex]
- [tex]\(2x - y = 7\)[/tex]:
[tex]\[ 2(1) - 5 = 7 \implies 2 - 5 = -3 \quad \text{(False)} \][/tex]
Based on the above checks, both equations [tex]\(x + y = 6\)[/tex] and [tex]\(2x + y = 7\)[/tex] pass through the point (1, 5). However, generally, we are asked for one specific line. Assuming the parallelism condition with considering only these specific equations alone, the matched solution is first in sequence fixing our problem parameters, and hence:
The equation representing the location of the second street is:
[tex]\[ x + y = 6 \][/tex]
