Math Problem Statement

Solve the given system of linear equations using the elimination method. Determine if the system is consistent or inconsistent and whether the equations are dependent or independent. The equations are: 3x = 12 - 2y and 6x = 45 - 11y.

Solution

Let’s solve the system of equations provided in the image step by step using the elimination method. The system is:

  1. 3x=12−2y3x = 12 - 2y
  2. 6x=45−11y6x = 45 - 11y

Step 1: Rearrange both equations into standard form

For 3x=12−2y3x = 12 - 2y, move 2y2y to the left: 3x+2y=123x + 2y = 12

For 6x=45−11y6x = 45 - 11y, move 11y11y to the left: 6x+11y=456x + 11y = 45

So the system becomes:

3x + 2y &= 12 \\ 6x + 11y &= 45 \end{aligned}$$ ### Step 2: Eliminate one variable To eliminate $$x$$, multiply the first equation by 2 so that the coefficients of $$x$$ match: $$2(3x + 2y) = 2(12) \implies 6x + 4y = 24$$ Now subtract the first equation ($$6x + 4y = 24$$) from the second equation ($$6x + 11y = 45$$): $$(6x + 11y) - (6x + 4y) = 45 - 24$$ $$7y = 21$$ Solve for $$y$$: $$y = 3$$ ### Step 3: Substitute $$y = 3$$ into one of the original equations Using the first equation ($$3x + 2y = 12$$): $$3x + 2(3) = 12$$ $$3x + 6 = 12$$ $$3x = 6$$ $$x = 2$$ ### Step 4: Check for consistency and dependency - There is one unique solution $$(x, y) = (2, 3)$$. - The system is **consistent** because a solution exists. - The system is **independent** because the two equations are not multiples of each other. ### Final Answer: - Select **A: There is one solution. The solution of the system is $$(2, 3)$$.** - Select **consistent** for the second part. - Select **independent** for the third part. Let me know if you have questions or need clarification! --- Here are 5 related questions to enhance your understanding: 1. How do you identify if a system is dependent, independent, or inconsistent? 2. What does it mean if a system has infinitely many solutions? 3. How do you solve systems of equations graphically, and how does it relate to algebraic solutions? 4. Why does multiplying an equation help in the elimination method? 5. How can substitution be used instead of elimination to solve this system? **Tip:** Always double-check your solution by substituting back into both original equations to confirm accuracy!

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Math Problem Analysis

Mathematical Concepts

Linear Equations
System of Equations
Elimination Method

Formulas

Standard form of linear equation: Ax + By = C
Elimination method: Combine equations to eliminate one variable

Theorems

Properties of equality
Definition of consistent and dependent systems

Suitable Grade Level

Grades 8-10

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