Question

Which of the following points is on the line \(y=7 x-4\) ? A. \((7,3)\) B. \((5,31)\) C. \((11,72)\) D. \((1,4)\)

Short answer

Point B \(5,31\) is on the line.

Step-by-step solution

- Determine the equation of the line

The given equation of the line is: \[ y = 7x - 4 \]

- Check point A

Substitute point A \(7, 3\) into the equation \[ y = 7x - 4 \] \[ 3 = 7(7) - 4 \] \[ 3 = 49 - 4 \] \[ 3 eq 45 \] Therefore, point A is not on the line.

- Check point B

Substitute point B \(5, 31\) into the equation \[ y = 7x - 4 \] \[ 31 = 7(5) - 4 \] \[ 31 = 35 - 4 \] \[ 31 = 31 \] Therefore, point B is on the line.

- Check point C

Substitute point C \(11, 72\) into the equation \[ y = 7x - 4 \] \[ 72 = 7(11) - 4 \] \[ 72 = 77 - 4 \] \[ 72 eq 73 \] Therefore, point C is not on the line.

- Check point D

Substitute point D \(1, 4\) into the equation \[ y = 7x - 4 \] \[ 4 = 7(1) - 4 \] \[ 4 = 7 - 4 \] \[ 4 eq 3 \] Therefore, point D is not on the line.

Key concepts

Linear Equations

Linear equations are mathematical expressions that create a straight line when graphed on a coordinate plane. The standard form of a linear equation in two variables, x and y, is given by: Where m is the slope of the line and b is the y-intercept. The slope represents the steepness of the line and the y-intercept is where the line crosses the y-axis. For example, in the equation: The slope is 7, and the y-intercept is -4. This means the line is steep and intersects the y-axis at -4.

Substitution Method

The substitution method is used to determine whether a point lies on a line represented by a linear equation. To use this method:

• Take the coordinate of the point (x, y)
• Substitute the x-value into the equation.
• Check if the resulting y-value matches the y-coordinate of the point.

If both sides of the equation are equal after substitution, the point lies on the line. In our example, point B (5,31) was substituted into the equation: Since both sides were equal, point B is on the line.

Coordinate Points

Coordinate points are pairs of numbers that represent a specific location on a coordinate plane. They are written in the form (x, y), where x represents the horizontal position and y represents the vertical position.

• The first number is the x-coordinate.
• The second number is the y-coordinate.

For example, the coordinate (5, 31) means moving 5 units along the x-axis and 31 units up the y-axis. In the exercise, we used the coordinates to check if points (7, 3), (5, 31), (11, 72), and (1, 4) are on the line.

Algebra

Algebra is a branch of mathematics focused on using symbols and letters to represent numbers and quantities in equations. It helps us solve problems by finding unknown values. To solve the given problem, we used algebraic techniques:

• We substituted the x-values into the linear equation.
• Performed arithmetic operations to simplify the expressions.
• Compared the results with the given y-values.

This method showed us which points, like point B (5, 31), satisfied the equation for that specific line.