Example Card: Elimination by Adding Equations
Solve systems of equations by the elimination method — add equations with opposite terms to cancel one variable and solve for the remaining unknown in Grade 9 Algebra.
Key Concepts
When two equations have perfectly opposite terms, adding them together can reveal the answer. This example uses the first key idea of this lesson, direct elimination.
Example Problem Solve the system by elimination and check the answer: $4x + 3y = 5$ and $ 4x + 5y = 11$.
Step by Step 1. The two equations have equal and opposite coefficients for the variable $x$, so we can add the equations to eliminate it. $$ \begin{array}{r} 4x + 3y = 5 \\ 4x + 5y = 11 \\ \hline 8y = 16 \end{array} $$ 2. Now, we solve for $y$. $$ y = 2 $$ 3. Substitute $2$ for $y$ in the first original equation to solve for $x$. $$ \begin{aligned} 4x + 3(2) &= 5 \\ 4x + 6 &= 5 \\ 4x &= 1 \\ x &= \frac{1}{4} \end{aligned} $$ 4. The solution is $( \frac{1}{4}, 2)$. 5. Check: Substitute $( \frac{1}{4}, 2)$ into the second equation to verify. $$ \begin{aligned} 4x + 5y &= 11 \\ 4( \frac{1}{4}) + 5(2) &\stackrel{?}{=} 11 \\ 1 + 10 &\stackrel{?}{=} 11 \\ 11 &= 11 \quad ✓ \end{aligned} $$ By adding the equations, the x terms cancelled out, making the problem much simpler. This additive elimination works whenever you spot opposite coefficients.
Common Questions
What is Example Card: Elimination by Adding Equations?
Example Card: Elimination by Adding Equations is a key concept in Grade 9 math. It involves applying specific rules and properties to simplify expressions, solve equations, or analyze mathematical relationships. Understanding this topic builds foundational skills needed for higher-level algebra and beyond.
How is Example Card: Elimination by Adding Equations used in real-world applications?
Example Card: Elimination by Adding Equations appears in practical contexts such as financial calculations, engineering problems, and data analysis. Mastering this skill helps students model and solve problems they will encounter in science, technology, and everyday decision-making situations.
What are common mistakes when working with Example Card: Elimination by Adding Equations?
Common errors include forgetting to apply rules to all terms, sign errors when working with negatives, and skipping verification steps. Always double-check by substituting answers back into the original problem and reviewing each algebraic step carefully.