Sure, let's work through the equation you've provided: \[ 7x - 6y = 18 \] Depending on what you're looking to find, there are a few different ways to approach this equation. Below are some common methods: ### 1. Solve for One Variable in Terms of the Other **a. Solving for $$ y $$ in terms of $$ x $$:** \[ 7x - 6y = 18 \] First, isolate the term containing $$ y $$: \[ -6y = -7x + 18 \] Now, divide both sides by $$ -6 $$ to solve for $$ y $$: \[ y = \frac{7}{6}x - 3 \] **b. Solving for $$ x $$ in terms of $$ y $$:** \[ 7x - 6y = 18 \] Isolate the term containing $$ x $$: \[ 7x = 6y + 18 \] Now, divide both sides by $$ 7 $$: \[ x = \frac{6}{7}y + \frac{18}{7} \] ### 2. Graphing the Equation To graph the equation $$ 7x - 6y = 18 $$, you can find two points that satisfy the equation and then draw a line through them. **a. When $$ x = 0 $$:** \[ 7(0) - 6y = 18 \Rightarrow -6y = 18 \Rightarrow y = -3 \] Point: $$ (0, -3) $$ **b. When $$ y = 0 $$:** \[ 7x - 6(0) = 18 \Rightarrow 7x = 18 \Rightarrow x = \frac{18}{7} \approx 2.57 \] Point: $$ \left(\frac{18}{7}, 0\right) $$ Plot these two points on a graph and draw a straight line through them. This line represents all the solutions to the equation $$ 7x - 6y = 18 $$. ### 3. Finding Integer Solutions If you're interested in finding integer solutions (where both $$ x $$ and $$ y $$ are integers), you can use the equation: \[ 7x - 6y = 18 \] Rearrange to solve for $$ y $$: \[ y = \frac{7}{6}x - 3 \] For $$ y $$ to be an integer, $$ \frac{7}{6}x $$ must result in an integer when subtracted by 3. This implies that $$ x $$ should be a multiple of 6. Let's set $$ x = 6k $$, where $$ k $$ is an integer. \[ y = \frac{7}{6}(6k) - 3 = 7k - 3 \] **Example Solutions:** - When $$ k = 0 $$: - $$ x = 0 $$ - $$ y = -3 $$ - Solution: $$ (0, -3) $$ - When $$ k = 1 $$: - $$ x = 6 $$ - $$ y = 4 $$ - Solution: $$ (6, 4) $$ - When $$ k = 2 $$: - $$ x = 12 $$ - $$ y = 11 $$ - Solution: $$ (12, 11) $$ And so on. You can generate as many integer solutions as needed by choosing different integer values for $$ k $$. ### 4. General Solution The equation $$ 7x - 6y = 18 $$ represents a linear relationship between $$ x $$ and $$ y $$. Depending on the context, you might use this relationship to model real-world scenarios, find specific solutions, or explore the properties of the equation further. If you have a specific goal in mind (like solving for a particular variable, finding solutions under certain conditions, or applying this equation to a real-world problem), please let me know, and I can provide a more targeted explanation! The equation $$ 7x - 6y = 18 $$ can be solved for one variable in terms of the other. For example, solving for $$ y $$ gives $$ y = \frac{7}{6}x - 3 $$.

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9) $ 7 x-6 y=18 $

Ask by Ball Ryan. in the United States

Jan 24,2025

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The equation $ 7x - 6y = 18 $ can be solved for one variable in terms of the other. For example, solving for $ y $ gives $ y = \frac{7}{6}x - 3 $.

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To solve the equation $ 7x - 6y = 18 $ for one variable, let's isolate $ y $: 1. Rearranging gives us $ -6y = -7x + 18 $. 2. Dividing through by -6 simplifies it to $ y = \frac{7}{6}x - 3 $. Now you can plug in any value of $ x $ to generate corresponding values for $ y $! Alternatively, if you want to find $ x $ given a specific $ y $, simply rearranging will do the trick! Just switch the roles and solve away!

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9) \( 7 x-6 y=18 \)

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