how to: graph, substitution, and elimination
y=3x+4
-12x+4y=16
-12 x + 4 (3x+4) = 17
-12 x + 12 x + 16 = 17
0 = 1 :)
inconsistent
These two lines are parallel and there is no solution
y = 3 x + 4 has m = slope = 3
4 y = 12 x + 16 is y = 3 x + 4
m = 3 again, same slope, different y intercept
To graph the equation y=3x+4, you can follow these steps:
Step 1: Create a coordinate plane by drawing two perpendicular lines, one horizontal (the x-axis) and one vertical (the y-axis), intersecting at the origin (0,0).
Step 2: Pick some values for x and substitute them into the equation to find the corresponding y-values. For example, choose x=-1, 0, and 1.
- When x=-1, substitute it into the equation: y=3(-1)+4 -> y=1
- When x=0, substitute it into the equation: y=3(0)+4 -> y=4
- When x=1, substitute it into the equation: y=3(1)+4 -> y=7
Step 3: Plot the points (x, y) obtained from the above substitutions on the coordinate plane. In this case, plot the points (-1, 1), (0, 4), and (1, 7).
Step 4: Connect the plotted points with a straight line. This line represents the graph of the equation y=3x+4.
Now, to solve the system of equations by substitution and elimination:
Substitution Method:
Step 1: Solve one equation for one variable in terms of the other variable. Let's solve the first equation, y=3x+4, for y: y-3x=4.
Step 2: Substitute this expression for y in the second equation, -12x+4y=16:
-12x + 4(3x+4) = 16
Step 3: Simplify and solve for x:
-12x + 12x + 16 = 16
16 = 16
Step 4: Since the equation above is always true, it means that the system is dependent or has infinitely many solutions.
Elimination Method:
Step 1: Multiply each equation by a suitable coefficient such that the coefficients of one of the variables will cancel out when the two equations are added or subtracted.
In this case, multiply the first equation by 4 and the second equation by 3 to make the coefficients of y cancel out:
4y = 12x + 16
12y = -36x - 48
Step 2: Add the two equations: (4y) + (12y) = (12x + 16) + (-36x - 48)
16y = -24x - 32
Step 3: Divide by 8 to simplify:
2y = -3x - 4
Step 4: Now, you have a simplified equation in terms of y and x. Rearrange the equation if necessary.
The solution to the system of equations using elimination is:
2y = -3x - 4
These methods can be used to graph equations and solve systems of equations using various techniques.