Problem 116 - College Algebra Essentials

Question

Use your graphing utility to enter each side of the equation separately under \(y_{1}\) and \(y_{2} .\) Then use the utility's TABLE or GRAPH feature to solve the equation. \(5 x+2(x-1)=3 x+10\)

Step-by-Step Solution

Answer

The solution for 'x' can be seen from the intersection point of the two graphed lines, which corresponds to the x-value solving the original equation.

Step 1: Rewrite Equation in 'y' Form

Modify the original equation into two equations resembling the 'y=mx+b' form. Let \(y_{1} = 5x + 2(x-1) = 7x - 2\) and \(y_{2} = 3x + 10\). 'y1' and 'y2' are the two equations to be graphed on a coordinate plane.

Step 2: Graph the Equations

Use a graphing tool to plot \(y_{1} = 7x - 2\) and \(y_{2} = 3x + 10\). These two lines represent the left and right sides of the original equation.

Step 3: Identify the Intersection Point

Observe that the two lines intersect at one point. This intersection point is the solution to the original equation.

Step 4: Read off the Solution

Read off the x-coordinate of the intersection point. This will be the solution for 'x' in the original equation.

Key Concepts

Using Graphing Calculators

Graphing calculators are powerful tools that make understanding algebra visual and interactive. They are particularly helpful when dealing with linear equations. To use a graphing calculator to solve an equation like the one given in our exercise, start by entering each side of the original equation as separate functions. This could be done by assigning them to different variables, such as y1 and y2. Most graphing calculators will have a function or feature labeled 'GRAPH' where these equations can be entered exactly as they would appear in your textbook.

Once input, use the calculator's graphing functionality to draw the lines represented by these equations on a coordinated plane. Many graphing calculators also offer a 'TABLE' feature, which will list out values of x and y for each function, letting you see the numerical relationship directly. Through these features, your calculator aids in visualizing the problem, which is an integral step for better comprehension.

Solving Algebraic Equations

Algebraic equations are the backbone of algebra and involve finding the value of unknowns that make the equation true. To solve an algebraic equation like the one in our example, 5x + 2(x-1) = 3x + 10, we need to isolate the variable, in this case, x. This usually involves combining like terms, distributing multiplications, and simplifying both sides of the equation. By rearranging the equation, one can often rewrite it in the form y = mx + b, where m is the slope of the line and b is the y-intercept. This form is particularly useful when graphing the equation on a coordinate plane or using a graphing calculator. Solving equations effectively is a crucial skill for not only algebra but for higher-level math and understanding the world through a quantitative lens.

Linear Equation Intersection

The point where two linear equations cross paths on a graph is known as the intersection. This intersection represents the exact value for which both linear equations yield the same result and hence, is the solution for the system of linear equations. In the context of the given problem, the intersection point of equations y1 = 7x - 2 and y2 = 3x + 10 can be found on the graph where the corresponding lines meet. The coordinates of this point give the solution for the variable x as well as the resulting value of y. When using a graphing calculator, this tool often has a built-in feature to compute this precise intersection, providing a fast and accurate solution to the problem.

Coordinate Plane Graphing

Graphing on a coordinate plane is a fundamental skill that transforms algebraic concepts from the abstract to the concrete. The coordinate plane consists of two perpendicular lines, called axes, which are used to plot points. The horizontal axis is typically labeled as the x-axis, while the vertical axis is the y-axis. When graphing a linear equation, each value of x corresponds with a value of y; plotting these ordered pairs (x, y) and then connecting them will reveal the visual representation of the equation, which is a straight line. In the given exercise, graphing the equations y1 and y2 allows us to visualize the relationships represented by the algebraic expressions and to directly identify the solution to the original problem by finding the intersection on the graph.