Question

LOGICAL REASONING In Exercises \(56-59\), tell whether the statement is true or false. Justify your answer. The \(x\) -intercept of the graph of \(3 x+5 y=30\) is 10

Short answer

The statement 'The x-intercept of the graph of \(3x+5y=30\) is 10' is true.

Step-by-step solution

Substitute y = 0 Into The Equation

Set \(y = 0\) in the equation \(3x + 5y = 30\) to find the x-intercept. This simplifies to \(3x = 30\).

Solve for x

Solve the equation to find the value of \(x\). This gives \(x = 30 / 3\) which simplifies to \(x = 10\).

Key concepts

Logical Reasoning

Logical reasoning is the process of using a structured, methodical approach to problem-solving. It involves taking the given information, breaking it down into smaller chunks, making inferences, and then arriving at a conclusion based on the data and logical principles.

For instance, in the exercise provided, the student is tasked with determining whether the statement \(x\)-intercept of the graph of \(3x + 5y = 30\) is 10 is true or false. This requires not only a knowledge of what an \(x\)-intercept represents but also the application of logical steps to validate or refute the statement. In essence, the student must logically deduce the position at which the line defined by the equation will intersect the \(x\)-axis when \(y\) equals zero.

Logical reasoning in mathematics often translates into identifying and applying the correct mathematical operations to arrive at a solution. Thus, a solid grasp of logical reasoning is indispensable for students to excel in various domains of math, including solving linear equations and graphing them accurately.

Solving Linear Equations

Solving linear equations is one of the fundamental skills in algebra. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can be written in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables.

To solve for a variable means to isolate the variable on one side of the equation and everything else on the other side. In our example, the equation \(3x + 5y = 30\) needs to be solved for \(x\) when \(y\) is 0. By substituting \(y\) with 0, we simplify the equation to \(3x = 30\), and by further dividing both sides by 3, we find that \(x = 10\). This value of \(x\) is the \(x\)-intercept of the graph of the equation.

Understanding step-by-step solutions to linear equations helps students tackle a wide array of problems. Substitution, elimination, and using the properties of equality are some methods employed in solving these types of equations, which form the foundation for more complex math concepts.

Graphing Linear Equations

Graphing linear equations is a visual representation of solutions to linear equations. It involves plotting a line on a coordinate plane, which provides a picture of all the possible solutions for the given equation. This line is constructed by finding the points at which the graph intersects the \(x\)-axis and \(y\)-axis, known as \(x\)-intercept and \(y\)-intercept, respectively.

In the context of our exercise, the \(x\)-intercept is a point where the graph crosses the \(x\)-axis, which means the value of \(y\) is zero at that point. By calculating this intercept, as shown in the step-by-step solution, we can begin to sketch the graph. Knowing the \(x\)-intercept is especially useful, as you only need one more point to draw a straight line, the simplest graph of a linear equation.

The \(x\)-intercept found in our example (\(x = 10\)) would be represented on a graph as the point \((10, 0)\). Graphing is not only a powerful tool for understanding linear equations better but also for solving real-world problems where visual interpretation is often necessary. By improving skills in graphing linear equations, students can visualize mathematical concepts, making abstract ideas more concrete and understandable.