Problem 52 - Algebra and Trigonometry

Question

Find all values of x satisfying the given conditions. $$y_{1}=7(3 x-2)+5, y_{2}=6(2 x-1)+24, \text { and } y_{1}=y_{2}$$

Step-by-Step Solution

Answer

The solution to the exercise is \(x = 3\).

Step 1: Simplifying y1 and y2

Simplify the two linear equations (y1 and y2):\n For y1: \(y_{1}=7(3x-2)+5 = 21x - 14 +5 = 21x - 9\) \n For y2: \(y_{2}=6(2x-1)+24 = 12x - 6 + 24 = 12x + 18\)

Step 2: Setting y1 equal to y2

Set \(y_{1}\) equal to \(y_{2}\) to find the intersection point:\n \(21x - 9 = 12x + 18\)

Step 3: Solving for x

Solve for x:\n Subtract 12x from both sides: \(9x - 9 = 18\)\n Add 9 to both sides: \(9x = 27\)\n Divide by 9: \(x = 3\)

Key Concepts

Understanding Linear Equations

In mathematics, a linear equation is an equation that plots a straight line when graphed. These equations are essential for modeling real-world relationships, which often showcase a constant rate of change. A standard form of a linear equation in two variables, such as x and y, is given by

• \( y = mx + b \)

Here, \(m\) represents the slope of the line, indicating how steep the line is, while \(b\) denotes the y-intercept or the point where the line crosses the y-axis. By analyzing linear equations, one can uncover valuable insights about variables and predict outcomes under varying conditions.
Linear equations are called so because every term is either a constant or the product of a constant with a single variable. Understanding the structure of these equations greatly assists in problem-solving by providing a clear picture of how variables interact.

Simplification Techniques

Simplification is a crucial step in solving equations as it makes the equation easier to interpret and solve. During simplification, one eliminates unnecessary parts of the equation to focus on the core components. Let's consider an example: in the problem at hand, we start with equations in the form of expressions, \(y_{1} = 7(3x-2)+5\) and \(y_{2} = 6(2x-1)+24\).
Here, simplification involves:

• Distributing multiplication over addition, i.e., multiplying terms inside the parentheses by the multiplication factor outside.
• Combining like terms to consolidate the expression into a simpler form.

After simplification, these equations become easier to handle: \(y_{1} = 21x - 9\) and \(y_{2} = 12x + 18\). By reducing complexity, simplification allows us to clearly see relationships and helps us progress to solving the equation more efficiently.

Intersection of Lines

The intersection of lines refers to the point where two lines cross each other on a graph. This concept is particularly useful in linear equations, where we might want to find the values of x and y that satisfy two different equations simultaneously. In the given problem, this is highlighted by the condition \(y_{1} = y_{2}\).
To find where two lines intersect:

• Set the equations equal to each other. This means you are finding the value of x for which both equations produce the same y-value.
• Resolve the equation to find the value of the variable, often x.

From our example, \(21x - 9 = 12x + 18\). By solving for x, we discover that the lines intersect when \(x = 3\). This tells us that, at this point, both lines share a common value, demonstrating their interception on the graph. Understanding the intersection helps in analyzing and predicting when different functions will yield equivalent results.