Solve System of Equations Using Elimination: 2x + 3y = 7 and -2x + 4y = 14

Math Problem Statement

Solve each system of equations without graphing and show your reasoning. The system is: 2x + 3y = 7 and -2x + 4y = 14.

Let's solve the given system of equations step by step.

The system of equations is:

1) \ 2x + 3y &= 7 \\ 2) \ -2x + 4y &= 14 \end{aligned}$$ ### Step 1: Add the equations To eliminate $$x$$, add both equations: $$(2x + 3y) + (-2x + 4y) = 7 + 14$$ This simplifies to: $$0x + 7y = 21 \quad \Rightarrow \quad 7y = 21$$ ### Step 2: Solve for $$y$$ Divide both sides by 7: $$y = \frac{21}{7} = 3$$ ### Step 3: Substitute $$y = 3$$ into one of the original equations Now substitute $$y = 3$$ into the first equation $$2x + 3y = 7$$: $$2x + 3(3) = 7 \quad \Rightarrow \quad 2x + 9 = 7$$ ### Step 4: Solve for $$x$$ Subtract 9 from both sides: $$2x = 7 - 9 \quad \Rightarrow \quad 2x = -2$$ Divide by 2: $$x = \frac{-2}{2} = -1$$ ### Solution The solution to the system is: $$x = -1, \quad y = 3$$ ### Verification Let's substitute $$x = -1$$ and $$y = 3$$ into the second equation to check: $$-2(-1) + 4(3) = 2 + 12 = 14$$ This is true, so the solution is correct. --- Would you like further details on this solution? Here are some related questions for practice: 1. What is the elimination method and when is it useful? 2. How would the solution differ if both equations were dependent? 3. What if one of the equations was non-linear—how would you approach it? 4. Could substitution work here, and how would you apply it? 5. How do you check if a system of equations has no solution or infinite solutions? **Tip:** Always verify your solutions by substituting them back into the original equations to avoid mistakes.

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Algebra
Systems of Equations
Elimination Method

Addition of equations to eliminate variables
Substitution of known values
Solving linear equations

Elimination Method
Linear Equations

Grades 8-10

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