Problem 23 - Algebra 1

Question

Solve the equation algebraically. Check your solution graphically. $$2 x+7=10$$

Step-by-Step Solution

Answer

The solution to the equation \(2x + 7 = 10\) is \(x = \frac{3}{2}\).

Step 1: Simplify the Equation

Start by subtracting 7 from both sides of the equation to isolate terms with x on one side. This transforms the equation to \(2x = 10 - 7\).

Step 2: Solve for x

Next calculate the result from the right side of the equation. This results in \(2x = 3\). Afterwards, divide both sides by 2 to get the value of \(x = \frac{3}{2}\).

Step 3: Check Solution Graphically

Plot the given equation and a horizontal line representing \(x = \frac{3}{2}\) on the same graph. If they intersect at \(x = \frac{3}{2}\), then the solution is correct.

Key Concepts

Algebraic Solutions

Algebraic solutions involve breaking down equations step-by-step to find the value of unknown variables. Let's take a closer look at this process using the equation \(2x + 7 = 10\).

• **Simplifying the Equation**: The first step in algebra is to simplify the equation by performing operations that help isolate the variable. Here, we subtract 7 from both sides to eliminate the constant term on the left, leading to \(2x = 3\).
• **Solving for the Variable**: With the simplified equation, \(2x = 3\), divide both sides by the coefficient of \(x\), which is 2. This leaves us with \(x = \frac{3}{2}\).
• **Consistency Check**: Perform a quick check to verify that the solution satisfies the original equation by substituting \(x = \frac{3}{2}\) back into \(2x + 7 = 10\) and confirming both sides equal 10 after calculation.

Algebraic solutions are a reliable way to find precise answers to linear equations, and this step-by-step approach often reveals the logical sequence behind the math.

Graphical Verification

Graphical verification is a fantastic tool to visually confirm the accuracy of an algebraic solution. It involves plotting the equation on a coordinate plane and checking for intersections.

• **Plot the Equation**: The equation \(2x + 7 = 10\) can be rewritten in terms of \(y\) as \(y = 2x + 7\) for graphing purposes. Plotting this line involves identifying points where \(x\) and \(y\) satisfy the equation.
• **Draw a Line for the Solution**: The next step is to draw a horizontal line at \(y = 3\), which represents our solution value that we derived, \(x = \frac{3}{2}\).
• **Find the Intersection**: The solution is validated if the lines intersect at \(x = \frac{3}{2}\). The crossing point provides a visual confirmation that the algebraic solution is indeed correct.

Graphical verification offers a powerful way to double-check results, particularly when visualizing how solutions fit within the broader graph of equations.

Solving Equations

Solving equations is a central skill in algebra that involves finding the values of variables that satisfy given conditions. Here's a breakdown of how you can approach solving linear equations like \(2x + 7 = 10\):

• **Understand the Equation Structure**: Linear equations, such as the one given here, typically consist of constants and a variable term. Recognizing this structure can help guide the solution approach.
• **Use Inverse Operations**: The key strategy is to perform inverse operations to dismantle the equation. For instance, addition can be countered with subtraction, and multiplication with division.By applying these operations systematically, you can isolate and find the value of \(x\).
• **Maintain Balance**: While manipulating the equation, it’s crucial to maintain balance – whatever you do to one side, you must do to the other. This ensures that the equation remains true.

Mastering the art of solving equations is essential as it lays the foundation for more complex math tasks, and provides insight into mathematical patterns and problem-solving techniques.