Rewrite the given equation in slope-intercept form and then graph the line. 7x + 5y – 35 = 0
What is the equation in slope-intercept form?
(Use integers or simplified fractions for any numbers in the equation.)
The Correct Answer and Explanation is :
To rewrite the given equation ( 7x + 5y – 35 = 0 ) into slope-intercept form, we need to solve for ( y ). The slope-intercept form of a linear equation is written as:
[
y = mx + b
]
where:
- ( m ) is the slope of the line, and
- ( b ) is the y-intercept.
Step-by-Step Solution:
- Start with the original equation:
[
7x + 5y – 35 = 0
] - Move all terms involving ( x ) and constants to one side by subtracting ( 7x ) and adding 35 to both sides:
[
5y = -7x + 35
] - Now, solve for ( y ) by dividing the entire equation by 5:
[
y = \frac{-7x + 35}{5}
]
This can be simplified by splitting the fraction:
[
y = -\frac{7}{5}x + 7
]
So, the equation in slope-intercept form is:
[
y = -\frac{7}{5}x + 7
]
Explanation:
- The slope ( m ) of the line is ( -\frac{7}{5} ), which means for every 5 units you move to the right (increasing ( x )), the value of ( y ) decreases by 7 units. This negative slope indicates that the line falls as you move from left to right.
- The y-intercept ( b ) is 7, meaning the line crosses the y-axis at the point ( (0, 7) ).
Graphing the Line:
To graph the equation ( y = -\frac{7}{5}x + 7 ), you can start by plotting the y-intercept at ( (0, 7) ). Then, use the slope of ( -\frac{7}{5} ) to plot another point. From ( (0, 7) ), move 5 units to the right (along the x-axis) and 7 units down (since the slope is negative). This gives you the point ( (5, 0) ). Drawing a line through these two points will give you the graph of the equation.
This slope-intercept form is useful because it directly shows the slope and y-intercept, making it easy to graph and analyze the line.