To find the equation of a line given in the form [tex]\( y = x + \_\_\_ \)[/tex], we need to determine the constant term that completes the equation.
Given:
[tex]\[ y = x + \_\_ \][/tex]
In general, the equation of a line in slope-intercept form is given by:
[tex]\[ y = mx + b \][/tex]
Here:
- [tex]\( m \)[/tex] is the slope of the line. In this case, it is 1, since the coefficient of [tex]\( x \)[/tex] is 1.
- [tex]\( b \)[/tex] is the y-intercept of the line, which is the value we need to find.
To find the y-intercept [tex]\( b \)[/tex], we need at least one point [tex]\((x_0, y_0)\)[/tex] through which the line passes. Suppose we are given a point (or we can assume one for demonstration):
- Let's assume a point [tex]\((x_0, y_0)\)[/tex].
For example, let's assume the point (0, 1):
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 1 \)[/tex].
Using this point in the equation [tex]\( y = x + b \)[/tex], we substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = 1 \)[/tex]:
[tex]\[ 1 = 0 + b \][/tex]
Thus:
[tex]\[ b = 1 \][/tex]
So, the equation of the line is:
[tex]\[ y = x + 1 \][/tex]
If we have a different point [tex]\((x_0, y_0)\)[/tex], we could repeat this process:
1. Substitute the point into the equation to solve for [tex]\( b \)[/tex].
2. Once [tex]\( b \)[/tex] is found, write the final equation [tex]\( y = x + b \)[/tex].
In summary, the missing constant term in the equation can generally be found if we have specific information about a point on the line or additional context to determine the value of [tex]\( b \)[/tex]. For this example, we assumed a point (0, 1) and found the equation:
[tex]\[ y = x + 1 \][/tex]
