find the solution the the system of equations by using either graphing or substitution
y = 2x - 1 and y = x + 3
We can solve the system by substitution.
First, we set the two equations equal to each other since they both equal y:
2x - 1 = x + 3
Next, we can solve for x:
2x - x = 3 + 1
x = 4
Now that we know x = 4, we can substitute it back into either of the original equations to solve for y. Let's use the first equation:
y = 2(4) - 1
y = 7
Therefore, the solution to the system of equations is (4, 7).
What will the graph look like for system of equations that has no solution the lines will be perpendicular lines will cross at one point. Both equations will form the same line or the lines will be parallel
The graph for a system of equations with no solution will consist of two lines that are parallel and never intersect. The lines will have the same slope but different y-intercepts.
To find the solution to the system of equations, you can use either the graphing method or the substitution method.
1. Graphing method:
To solve the system using graphing, you will plot the two equations on a coordinate plane and identify the point where they intersect.
Start by plotting the first equation, y = 2x - 1:
- Choose several values for x and use the equation to calculate the corresponding y-values.
- Plot these points on the coordinate plane.
- Draw a straight line passing through these points.
Next, plot the second equation, y = x + 3, on the same coordinate plane, using the same method.
- Choose several values for x and calculate the corresponding y-values.
- Plot these points on the same coordinate plane.
- Draw a straight line passing through these points.
The solution to the system of equations is the point where the two lines intersect. Read the coordinates of this intersection point to find the solution.
2. Substitution method:
To solve the system using substitution, you will substitute one equation into the other to eliminate one variable and then solve for the remaining variable.
Start with the first equation, y = 2x - 1:
- Solve this equation for y: y = 2x - 1.
Next, substitute this expression for y into the second equation, y = x + 3:
- Replace y in the second equation with 2x - 1: 2x - 1 = x + 3.
Now, solve this equation for x:
- Subtract x from both sides: 2x - x - 1 = 3.
- Simplify: x - 1 = 3.
- Add 1 to both sides: x - 1 + 1 = 3 + 1.
- Simplify: x = 4.
Now that you have the value of x, substitute it back into one of the original equations to solve for y. Let's use the first equation:
- Substitute x = 4 into y = 2x - 1: y = 2(4) - 1.
- Simplify: y = 8 - 1.
- Solve: y = 7.
Therefore, the solution to the system of equations is x = 4 and y = 7.