Problem 33 - Algebra 1

Question

Use the substitution method to solve the linear system. $$\begin{aligned} &7 g+h=-2\\\ &g-2 h=9 \end{aligned}$$

Step-by-Step Solution

Answer

The solution to the system of equations is \(g = 1\), \(h = -13/3\).

Step 1: Express one variable in terms of the other

From the second equation \(g-2 h=9\), express \(g\) in terms of \(h\). That means, isolate \(g\), hence \(g = 9 + 2h\)

Step 2: Substitute the expression into the other equation

Substitute \(g = 9 + 2h\) from step 1 into the first equation which is \(7g + h = -2\). This gives: \(7(9 + 2h) + h = -2\), simplify this to give \(63 + 14h + h = -2\) which results in \(15h = -65\)

Step 3: Solve for the first variable

Solve for \(h\) in step 2, you get \(h = -65/15 = -13/3\).

Step 4: Solve for the second variable

Substitute \(h = -13/3\) into the equation found in step 1, which is \(g = 9 + 2h\), to get \(g = 9 + 2(-13/3) = 9 - 26/3 = 1\).

Key Concepts

Linear Equations

Linear equations are mathematical expressions that form straight lines when graphed. They involve variables raised to the power of one, and their general form is \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. These equations are fundamental for describing relationships in algebra and are the building blocks for solving complex problems. In our problem set, we have the equations \(7g + h = -2\) and \(g - 2h = 9\). Understanding linear equations is crucial because they help in determining relationships where one quantity is dependent on another. For instance, in our case, the variables \(g\) and \(h\) could represent two different measured quantities that have a certain relationship expressed through the equations. Recognizing their linear nature allows us to apply various strategies, such as graphical methods, elimination, and substitution, to solve them.

Solving Systems of Equations

When dealing with equations involving two or more variables, we often face systems of equations. This means more than one equation needs to be solved simultaneously. Our goal with systems of equations is to find a common solution set that satisfies all the equations in the system.In the given example, we are solving for the values of \(g\) and \(h\) that make both equations true at the same time. The two main methods used include:

• Substitution: Using one equation to express a variable in terms of others and replacing that expression in another equation.
• Elimination: Adding or subtracting equations to eliminate one variable and making it simpler to solve for another.

Both methods aim to do the same thing: find the intersection points on the graph where the lines represented by the equations meet.For the equations \(7g + h = -2\) and \(g - 2h = 9\), understanding these techniques means you can comprehensively determine \(g\) and \(h\) from their relationships.

Substitution Technique

The substitution technique is a strategic way to solve systems of equations, where you solve one of the equations for one variable and substitute that expression into the other equation. This method is efficient for systems where one variable can be readily isolated.Let's explore it with our example:1. **Isolate a variable:** Start by taking the simpler equation, \(g - 2h = 9\), and solve for \(g\). This gives us \(g = 9 + 2h\).2. **Substitute the expression:** Replace \(g\) in the first equation \(7g + h = -2\) with the expression from step 1, resulting in the statement \(7(9 + 2h) + h = -2\).3. **Solve for one variable:** Simplify the equation to find \(h\), in this case, \(15h = -65\), leading to \(h = -13/3\).4. **Back-substitute for the other variable:** Using the value of \(h\), substitute back to find \(g\), giving \(g = 1\).Many students find substitution intuitive as it systematically reduces the number of unknowns without altering the equation's essence. This clarity makes the substitution method a popular choice for tackling linear systems.