Problem 33 - Algebra 1

Question

Solve the equation graphically. Check your solution algebraically. $$7-9 x=-11$$

Step-by-Step Solution

Answer

The solution to the equation \(7 - 9x = -11\) is \(x = 4/9\).

Step 1: Simplify the Equation

First, rearrange the equation in the form 'expression = 0' by adding \(9x\) to both sides. We obtain the equation \(9x + 7 = 11\).

Step 2: Solve for x

Next, subtract 7 from both sides to isolate \(x\) and yield the equation \(9x = 4\). Then, divide both sides by 9 to find the value of \(x\), which comes out to be \(x = 4/9\).

Step 3: Graphical Representation

Plot the line of the equation \(9x + 7 - 11 = 0\) on a graph. The x-intercept of this line occurs at \(x = 4/9\), which is our solution.

Step 4: Verification

Finally, to check the solution, substitute \(x=4/9\) back into the original equation and ensure that both sides balance. Doing this, \(7 - 9*(4/9)\) simplifies to \( -4 \) as expected.

Key Concepts

Graphical Solution

When solving a linear equation graphically, we visualize the equation as a straight line on a coordinate plane. In simplest terms, each equation corresponds to a line, and solving graphically involves finding the x-intercept. This is the point where the line crosses the x-axis.
To solve the equation \( 7 - 9x = -11 \) graphically, we first transform it into the standard linear form \( y = mx + b \). The graphical solution involves plotting this line and observing where it intersects the x-axis, which represents the solution to the equation.
The transformation leads to the equation \( 9x + 7 - 11 = 0 \). We plot this equation on a graph with x and y axes. The x-intercept, where the y-value is zero, indicates the solution \( x = 4/9 \). This visualization helps us see how changes in x affect y, making complex or abstract algebraic principles more tangible.

Algebraic Verification

Checking your solution algebraically is a crucial step to ensure the correctness of your graphical solution. This process involves substituting the solution back into the original equation to verify that both sides balance, confirming the solution is accurate.
For the equation \( 7 - 9x = -11 \), once we find the solution \( x = 4/9 \) graphically, we can't always stop there. We need to plug \( x = 4/9 \) back into the equation to ensure no errors were made during solving or plotting. Perform this step by calculating \( 7 - 9(4/9) \). If solved correctly, the original equation balances: the left-hand side simplifies to \(-4 \) which matches the right-hand side of the equation, confirming that our graphical solution is indeed correct.

Linear Equations

Linear equations form the backbone of algebra and involve expressions where each term is either a constant or the product of a constant and a single variable. The general form is \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable.
The solution to a linear equation is the value of the variable that makes the equation true. Thus, solving linear equations means finding this value. We use different methods, such as graphing, substitution, or elimination, to isolate the variable and determine its exact value.
In our problem, the equation \( 7 - 9x = -11 \) is a classic example of a linear equation. Solving it often involves rearranging terms and using inverse operations to isolate \( x \). The symmetry and predictability of linear equations make them a fundamental starting point for understanding more complex algebraic concepts.