Solve for x.
3x = 6x - 2
Answer (2)
Solve the first system of equations by substitution and find x = 5 6 and y = 5 2 .
Indicate that the solution to the first system of inequalities is found by graphing.
Solve the second system of equations by substitution and find x = 2 and y = 5 .
Indicate that the solution to the second system of inequalities is found by graphing.
Explanation
Problem Overview We are given two systems of equations and two systems of inequalities. We need to solve them.
Solving the First System of Equations First, let's solve the first system of equations:
y = − 3 x + 4 y = 2 x − 2
Setting the two equations equal to each other:
− 3 x + 4 = 2 x − 2
Adding 3 x to both sides:
4 = 5 x − 2
Adding 2 to both sides:
6 = 5 x
Dividing by 5 :
x = 5 6
Now, substitute x = 5 6 into the first equation:
y = − 3 ( 5 6 ) + 4
y = − 5 18 + 5 20
y = 5 2
Solving the First System of Inequalities Next, let's solve the second system of inequalities:
\frac{1}{2}x - 1"> y > 2 1 x − 1 2 x + 3 y ≤ 6
To solve this system, we would typically graph the inequalities and find the region that satisfies both.
Solving the Second System of Equations Now, let's solve the first system of equations in the Exit Ticket:
y = 2 x + 1 y = − 2 1 x + 6
Setting the two equations equal to each other:
2 x + 1 = − 2 1 x + 6
Adding 2 1 x to both sides:
2 5 x + 1 = 6
Subtracting 1 from both sides:
2 5 x = 5
Multiplying by 5 2 :
x = 2
Now, substitute x = 2 into the first equation:
y = 2 ( 2 ) + 1
y = 4 + 1
y = 5
Solving the Second System of Inequalities Finally, let's solve the second system of inequalities in the Exit Ticket:
2x - 5"> y > 2 x − 5 3 x + 4 y ≤ 12
To solve this system, we would typically graph the inequalities and find the region that satisfies both.
Final Answer The solution to the first system of equations is x = 5 6 and y = 5 2 . The solution to the first system of inequalities is the region that satisfies both \frac{1}{2}x - 1"> y > 2 1 x − 1 and 2 x + 3 y ≤ 6 . The solution to the second system of equations is x = 2 and y = 5 . The solution to the second system of inequalities is the region that satisfies both 2x - 5"> y > 2 x − 5 and 3 x + 4 y ≤ 12 .
Examples
Systems of equations and inequalities are used in various real-world applications, such as determining the optimal mix of products to maximize profit, designing structures that can withstand certain loads, and modeling the spread of diseases. For example, a company might use a system of equations to determine the number of units of each product to produce in order to maximize profit, given constraints on resources such as labor and materials. Similarly, engineers might use systems of inequalities to design bridges that can withstand certain loads, given constraints on the materials used and the cost of construction. Understanding how to solve systems of equations and inequalities is therefore essential for many fields.
To solve the equation 3 x = 6 x − 2 , we first isolate x by subtracting 6 x from both sides, resulting in − 3 x = − 2 . Dividing both sides by -3 gives us the solution x = 3 2 . Thus, the value of x is 3 2 .
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