Match the system of equations to the number of solutions it has
.y = 3x + 6
y = -13x - 4
The system of equations you provided represents two separate lines.
The first equation, y = 3x + 6, represents a line with a slope of 3 and a y-intercept of 6.
The second equation, y = -13x - 4, represents a line with a slope of -13 and a y-intercept of -4.
Since these lines have different slopes, they will intersect at exactly one point. Thus, the system of equations has one solution.
To determine the number of solutions for a system of equations, we need to analyze the slopes and y-intercepts of the equations.
The given system of equations is:
1. y = 3x + 6
2. y = -13x - 4
To simplify the process, let's rewrite the equations in slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept.
1. y = 3x + 6
This equation has a slope (m) of 3 and a y-intercept (b) of 6.
2. y = -13x - 4
This equation has a slope (m) of -13 and a y-intercept (b) of -4.
Now let's examine the slopes:
- If the slopes are equal, the system has infinitely many solutions. The lines are superimposed and coincide.
- If the slopes are different, the system has a unique solution. The lines intersect at a single point.
- If the slopes are equal and the y-intercepts are different, the system has no solution. The lines are parallel and will never intersect.
Comparing the slopes of the given equations:
The slope of the first equation, 3, is different from the slope of the second equation, -13.
Therefore, the given system of equations has a unique solution.
To determine the number of solutions a system of equations has, we can compare the slopes (coefficients of x) of both equations.
The given system of equations is:
1) y = 3x + 6
2) y = -13x - 4
Since both equations are in slope-intercept form (y = mx + b), we can directly compare the slopes.
The slope of the first equation (equation 1) is 3, and the slope of the second equation (equation 2) is -13.
Because the slopes are different, the system of equations has exactly 1 solution.
Therefore, the matching number of solutions for this system of equations is 1.