x+3y>-3 and 4x+y=4
To solve the system of linear inequalities x + 3y > -3 and 4x + y = 4, we can use a combination of substitution and elimination methods. Here's the step-by-step process:
Step 1: Solve one equation for one variable in terms of the other variable.
From the second equation, we can solve for y in terms of x:
y = 4 - 4x
Step 2: Substitute the expression for y into the other equation.
Substitute (4 - 4x) for y in the first equation:
x + 3(4 - 4x) > -3
Step 3: Simplify and solve for x.
Distribute:
x + 12 - 12x > -3
-11x + 12 > -3
-11x > -3 - 12
-11x > -15
Divide by -11 and remember to reverse the inequality sign when dividing by a negative number:
x < -15/-11
x < 15/11
So, x is less than 15/11.
Step 4: Substitute the value of x into the equation y = 4 - 4x to find y.
Substitute x = 15/11 into y = 4 - 4x:
y = 4 - 4(15/11)
y = 4 - 60/11
y = 44/11 - 60/11
y = -16/11
So, y is -16/11.
Therefore, the solution to the system of linear inequalities x + 3y > -3 and 4x + y = 4 is:
x < 15/11
y = -16/11
To find the solution to the system of equations:
1. Rewrite the equations in slope-intercept form (y = mx + b):
- The first equation: x + 3y > -3
Subtract x from both sides to isolate 3y:
3y > -x - 3
Divide by 3 to solve for y:
y > -x/3 - 1
- The second equation: 4x + y = 4
Subtract 4x from both sides to isolate y:
y = -4x + 4
2. Graph the two equations on a coordinate plane:
- Plot the points of both lines.
- The first line (y > -x/3 - 1) is a dashed line, as the inequality is greater than.
- The second line (y = -4x + 4) is a solid line.
3. Determine the overlapping region on the graph:
- Shaded area: The area where the dashed line and solid line intersect.
- This region represents the solution to the given system of equations.
4. Write the solution in interval notation:
- The solution is the set of all points within the shaded region on the graph.
Please note that without further constraints or instructions, this is a visual representation of the solution.